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ESCC Mathematics Tutorials

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Mathematics Tutorials

These pages are intended to aide in the preparation for the Mathematics Placement test.

They are not intended to be a substitute for any mathematics course.

Arithmetic Tutorials Algebra I Tutorials Algebra II Tutorials

Word Problems

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Arithmetic Tutorials


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Arithmetic Tutorials

Whole Numbers Fractions Sets of Numbers Least Common Multiple Properties of Real Numbers Greatest Common Factors Addition of Whole Numbers Reducing Fractions Subtraction of Whole Numbers Addition of Fractions and Mixed Numbers Multiplication of Whole Numbers Subtraction of Fractions and Mixed Numbers Division of Whole Numbers Multiplication of Fractions and Mixed Numbers Order of Operations Division of Fractions and Mixed Numbers Exponential Notation Converting between Mixed Numbers and Improper Fractions Prime Numbers and Factoring Roman Numerals

Decimals Ratio and Proportions Introduction to Decimals Introduction to Ratios Addition of Decimals Introduction to Rates Subtraction of Decimals Introduction to Proportions Multiplication of Decimals Division of Decimals Converting Fractions to Decimals Scientific Notation

Percents Signed Numbers Introduction to Percents Negative Numbers Solving Percent Equations Addition and Subtraction of Signed Numbers

Multiplication and Division of Signed Numbers

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Algebra I Tutorials


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Algebra I Tutorials

Linear Equations and Inequalities What is a Variable? Evaluating Algebraic Expressions Combining Like Terms in Algebraic Expressions Properties of Equalities Algebraic Equations Solving First Degree(Linear) Equations Literal Equations and Formulas Linear Inequalities

Exponents Factoring Polynomials Multiplication with Exponents Common Factors Division with Exponents Factoring using Common Factors

Polynomials Factoring by Grouping

What is a Polynomial? Solving Equations by Factoring Evaluating a Polynomial Factoring the difference of two squares Addition of Polynomials Factoring the sum or difference of cubes Subtraction of Polynomials Multiplication of Polynomials Special Products Division of Polynomials

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Algebra II Tutorials


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Algebra II Tutorials

Rational Expressions Introduction to Rational Expressions Equivalent Rational Expressions Simplifying Rational Expressions Addition and Subtraction of Rational Expressions Multiplication and Division of Rational Expressions Complex Fractions Equations involving Fractions

Quadratic Equations Introduction to Quadratic Equations

Solving Quadratic Equations Factoring Extracting the Root Completing the Square Quadratic Formula

Complex Numbers Introduction to Complex Numbers Addition and Subtraction of Complex Numbers Multiplication of Complex Numbers Division of Complex Numbers Complex Solutions to Quadratic Equations

Linear Equations Graphing Linear Equations

Cartesian Coordinate System Graphs of Linear Equations Intercepts

Slope of the Line

Systems of Linear Equations Introduction to Systems of Linear Equations Solving by Substitution

Solving by Elimination

Roots and Radicals Common Roots and Radicals Properties of Radicals Addition and Subtraction of Radicals Multiplication of Radicals Division of Radicals Equations involving Radicals

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Word Problems


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Word Problems

Translating Word Problems to Algebra Number Problems

Age Problems Coin Problems Work Problems

Mixture Problems Distance Problems

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Mathematical Numbers

Natural Numbers

Natural numbers, also known as counting numbers, are the numbers beginning with 1, with each successive number greater than its predecessor by 1. If the set of natural numbers is denoted by N, then

N = { 1, 2, 3, ......}

Whole Numbers

Whole numbers are the numbers beginning with 0, with each successive number greater than its predecessor by 1. It combines the set of natural numbers and the number 0. If the set of whole numbers is denoted by W, then

W= { 0, 1, 2, 3, .......}

Rational and Irrational Numbers

Rational numbers are the numbers that can be represented as the quotient of two integers p and q, where q is not equal to zero. If the set of rational numbers is denoted by Q , then

Q = { all x, where x = p / q , p and q are integers, q is not zero}

Rational numbers can be represented as:

(1) Integers: (4 / 2) = 2, (12 / 4) = 3 (2) Fractions: 3 / 4, 13 / 3 (3) Terminating Decimals: (3 / 4) = 0.75, (6 / 5) = 1.2 (4) Repeating Decimals: (13 / 3) = 4.333... (4 / 11) = .363636...

Conversely, irrational numbers are the numbers that cannot be represented as the quotient of two integers, i.e., irrational numbers cannot be rational numbers and vice-versa. If the set of irrational numbers is denoted by H, then

H = { all x, where there exists no integers p and q such that x = p / q, q is not zero }

Typical examples of irrational numbers are the numbers and e, as well as the principal roots of rational numbers. They can be expressed as non-repeating decimals, i.e., the numbers after the decimal point do not repeat their pattern.

Real Numbers

Real numbers are the numbers that are either rational or irrational, i.e., the set of real numbers is the union of the sets Q and H. If the set of real numbers is denoted by R , then

R = Q H

Since Q and H are mutually exclusive sets, any member of R is also a member of only one of the sets Q and H. Therefore; a real number is either rational or irrational (but not both). If a real number is rational, it can be expressed as an integer, as the quotient of two integers, and a terminating or repeating decimal can represent it; otherwise, it is irrational and cannot be represented in the above formats.

Complex Numbers

Complex numbers are the numbers with the format a + b i, where a and b are real numbers and i = - 1. If we denote the set of complex numbers by C, then

C = { a + b i , where a and b are real numbers, i = -1 }

If in the number x = a + b i, b is set to zero, then x = a, where a is a real number. Thus, all real numbers are complex numbers, i.e., the set of complex numbers includes the set of real numbers.

Real Number System

The real number system is comprised of the set of real numbers and the arithmetic operations of addition and multiplication (subtraction, division and other operations are derived from these two). The rules and relationships that govern the real number system are the basis for most algebraic manipulations.

Properties of Real Numbers

All real numbers have the following properties:

(1) Reflexive Property

For any real number a, a = a. Example: 3 = 3, y = y, x + z = x + z (x, y and z are real numbers)

(2) Symmetric Property

For any real numbers a and b, if a = b, then b = a. Example: If 3 = 1 + 2, then 1 + 2 = 3

(3) Transitive Property

For any real numbers a, b and c, if a = b and b = c, then a = c. Example: If 2 + 3 = 5 and 5 = 1 + 4, then 2 + 3 = 1 + 4.

(4) Substitution Property

For any real numbers a and b, if a = b, then a may be replaced by b, and b may be replaced by a, in any mathematical statement without changing the meaning of the statement.

Example: If a = 3 and a + b = 5, then 3 + b = 5.

(5) Trichotomy Property

For any real numbers a and b, one and only one of the following conditions holds:

(1) a is greater than b ( a > b) (2) a is equal to b ( a = b) (3) a is less than b ( a < b)

Example: 3 < 4 , 4 + 2 = 6 , 7 > 5

Absolute Values

The absolute value of a real number is the distance between its corresponding point on the number line and the number 0. The absolute value of the real number a is denoted by |a|.

From the diagram, it is clear that the absolute value of nonnegative numbers is the number itself, while the absolute value of negative integers is the negative of the number. Thus, the absolute value of a real number can be defined as follows:

For all real numbers a, (1) If a >= 0, then |a| = a. (2) If a < 0, then |a| = -a.

Examples: | 2 | = 2 | -4.5 | = 4.5 | 0 | = 0

Addition of Whole Numbers

Addition is the process of finding the total of two or more numbers.

We first learn addition through counting

*** + **** = ******* 3 4 7

We can also look at addition on our old friend the number line.

each of these line segments is 3 units long (length does not depend upon position 7(total)

3 + 4

Therefore, we also have 3 + 4 = 7

There a few special properties of addition that we need to be aware of.

Addition Property of Zero Zero added to any number does not change the number 4 + 0 = 4 0 + 1 = 1

Commutative Property of Addition Two numbers can be added in any order, the sum is the same 4 + 8 = 12 8 + 4 = 12

Associative Property of Addition Grouping of addition in any order does not change the sum (4 + 2) + 3 = 6 + 3 = 9 4 + (2 + 3) = 4 + 5 = 9

The number line and other assorted aides are fine to learn with but the basic addition of 2 one-digit numbers must be memorized. The addition of larger numbers is basically the repeated usage of the basic addition facts.

Addition Table + 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18

To use this table to add two numbers, find the first number to be added in the top row (put your finger there). Find the second number to be added in the first column (put another finger there). Now, bring your fingers together (first finger straight down and second finger straight across). The place where your fingers meet is the sum.

Addition of larger numbers The easiest way to add larger numbers is to arrange the numbers vertically, keeping the digits of the same place value in the same column.

Add 321 + 6472

321 +6472 6793

Addition with carry If one of the columns in an addition sums to a value greater than 9 then you must

perform a carry. Write down the ones digit of the sum and carry the tens digit into the next column to the left.

Add 645 + 476 645

+476 5 + 6 = 11

1 645

+476 1 + 4 + 7 = 12 1

645 +476 1 + 6 + 4 = 11 1121

1

When do we use Addition? There are several types of problems that require the use of addition. One of the

major clues to the use of addition is the major key words leading to addition.

Addition Key Words Added to 3 added to 5 3 + 5 More than 7 more than 5 5 + 7 The sum of the sum of 3 and 9 3 + 9 Increased by 4 increased by 6 4 + 6 The total of the total of 3 and 8 3 + 8 Plus 5 plus 10 5 + 10

Subtraction of Whole Numbers

Subtraction is the process of finding the difference between two numbers.

We learn subtraction (as with addition) by counting.

******** - ******** = ***** 8 3 5

Minuend Subtrahend Difference

We can also show subtraction on the familiar number line.

8

3 5

You can readily see that addition and subtraction are related Subtrahend +Difference = Minuend

3 + 5 = 8

You can use this fact to check you subtraction with addition.

Subtraction of Larger Numbers

To perform subtraction on larger numbers by arranging the numbers vertically ( as in addition). Then subtract the numbers in each column.

Subtract 8955 2432

8955 -2432 6523

Subtraction with borrowing

If during the course of perform a subtraction on a large number, you are attempting to subtract a large number from a smaller number you must use borrowing.

Subtract 692 378

692 -378 you can not subtract 8 from 2 so we need to borrow 1 ten from the 9 tens in the

tens column leaving 8 tens and 12 ones.

81 692

-378 12 8 = 4 4

81 692

-378 8 7 = 1 14

81 692

-378 6 3 = 3 314

When do we use Subtraction? There are many key words that lead us to perform subtraction.

Subtraction Key Words

Minus 8 minus 5 8 5 Less 9 less 3 9 3 Less than 2 less than 7 7 2 The difference between the difference between 8 and 2 8 2 Decreased by 5 decreased by 1 5 1

Multiplying Whole Numbers

Multiplication is basically repeated additions.

3 2 = 2 + 2 + 2 = 6

6 8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 48

The numbers that are multiplied are called factors (6 and 8) and the result is the product (48).

There are three basic ways to represent multiplication a b a b all mean the same thing (a multiplied by b) a(b)

As is addition the best way to learn multiplication is to memorize the basic facts.

Multiplication Table 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81

To use the table, place one finger on the top row on the first factor and place another finger on the second factor on the first column. Bring the finger together

and you have the product.

Properties of Multiplication There are several useful properties of multiplication that will help us in our computations

Multiplication Property of Zero The product of any number and 0 is 0

04 = 0 70 = 0

Multiplication Property of One The product of any number and one is the number

15 = 5 61 = 6

Commutative Property of Multiplication Two numbers can be multiplied in either order and the product is unchanged.

43 = 34 = 12

Associative Property of Multiplication Grouping the numbers in a multiplication problem in any order gives the same

result (42)3 = 83 = 24 4 (23) = 46 = 24

Multiplying Larger Numbers Multiplying large numbers involves the repeated usage of basic one-digit

multiplication facts.

Multiply 37 4

3 7 4 47=28 write the 8 in the ones column and carry the 8

above the tens column

2 3 7

4 8

34=12, add the carry digit 12 + 2 = 14

3 7 4

148

Multiply 47 23

47 23 47 3 = 141

47 23

141 2047 = 940

47 23

141 940

1081

141 + 940 = 1081

Multiply 439 206

439 203 3 439 = 2634

439 203 2634

0 439 = 000

439 203 2634 0000

200 x 439 = 87800

439 203 2634

00000 87800 90434

2634 + 0000 + 87800 = 90434

When do we use multiplication? There are key words that indicate the use of multiplication

Multiplication Key Word Times 7 times 3 73

The product of The product of 6 and 9 69 Multiplied by 8 multiplied by 2 82

Division of Whole Numbers

Division is used to separate objects into groups of equal size.

Division is the inverse of multiplication (34 = 12 and 12 3 = 4 and 12 4 = 3)

We right division in two different ways 12 4 is the same as 4 12

6 Look at the division 4 24 , we refer to 4 as the divisor, 24 as the dividend and 6 as the

quotient. In general, we have quotient

divisor dividend

6 Also, we can see the relationship between division and multiplication 4 24

because 4 6 = 24

Important Division Rules

Any number, except zero, divided by itself equals 1 1 1

8 8 2 2

Any number divided by 1 is the number itself 8 27

1 8 1 27

Zero divided by any number is zero 0 0

7 0 101 0

Division by zero is not allowed ?

0 8 , there is no number whose product with 8 is 0

Dividing single digits into larger numbers

Divide 3192 4

4 3192 4 divides into 31 - 7 times since 4 * 7 = 28

39 28

7 4 3192 subtract 28 from 31, bring down the 9

3 36 39

28

79 4 3192

3 36 39

28

7 4 3192

4 divides into 39 - 9 times since 4*9=36, subtract 39-36

7 4 3192

28 bring down the 2,

32 36 39

4 divides into 32-8 times since 4*8=32, subtract 32 32 = 0 32 0

Dividing by single digit with a remainder

Divide 3 14

4 3 14

3*4 = 12

2 12

4 3 14

subtract 14 12 = 2

2 12

4 2 3 14

r so the result is 4 with a remainder of 2 (3*4 + 2=14)

Divide larger numbers

Divide 34 1598

34 1598 Think about 3 * 5 = 15 but 5 * 34 = 170, which is larger than 159, so use 4

4 34 1598 4*34 = 136

subtract 159 136 = 23 bring down the 8

238 136

4 34 1598

7*34 = 238 subtract 238-238= 0

238 238

136

47 34 1598

So the solution is 37 since 47 * 34 = 1598

When do we use Division? There are a couple of key words that indicate the use of division.

Division Key Words The quotient of The quotient of 9 and 3 9 3

Divided by 6 divided by 2 6 2

Order of Operations

Many times, in math classes, the problems involve more than one operation in the same problem. We need a system to determine the order in which we perform our operations. There is a hierarchy of operations that keep us from being confused by the messy problems.

The Order of Operations can be remembered by learning the phrase Please Excuse My Dear Aunt Sally.

Parenthesis Exponents Multiplication & Division Addition & Subtraction (4 levels)

Ex. 4+5*6 from left to right we see addition and multiplication

(multiplication first priority) 4+30 now we can add 34

Ex. 55 2*10 + 43 subtraction multiplication addition

(multiplication first) 55 20 + 43 now left to right (subtraction and division equal

priority) 35 + 43 78

Ex. (3 + 4)2 parenthesis and exponent (p first) 72 now exponent 49

Exponential Notation

Repeated multiplication of the same number can be written in two different ways

3*3*3*3 or 34 exponent

The exponent shows how many time 3 is multiplied by itself. 34 is in a format called exponential notation.

Examples of exponential notation

6 = 61 six to the first power (usually dont write the 1) 6*6 = 62 six squared or six to the second power 6*6*6 = 63 six cubed or six to the third power etc.

3*3*3*3*5*5*5 = 3 4*53

Place values are actually powers of 10

Ten 101

Hundred 102

Thousand 103

Ten-Thousand 104

Hundred-Thousand 105

Million 106

Factoring Numbers

We can divide whole numbers into two categories (prime and composite). Prime numbers are numbers that are only divisible by 1 and itself such as 3, 5, 11, 13. Composite numbers are numbers that are products of prime numbers such as 6, 15, 20.

One of the major things that we need to do with whole numbers is to factor the composite numbers into their prime parts, called factoring.

Ex. 10 = 2*5 20 = 2*2*5

Ex. Factor 105 Start with the small primes and check for divisibility 2 does not work since 2 does not divide 105 evenly but 3 works 105 = 3*35 now factor 35 as 5*7 so we get 105 = 3*5*7

Ex. Factor 129 2 wont work but 3 does 129 = 3*43, 43 is prime so 129 = 3 *43

Ex. Factor 400 400 = 2*200 FACTOR 200 400 = 2*2*100 FACTOR 100 400 = 2*2*2*50 FACTOR 50 400 = 2*2*2*5*5 DONE

Roman Numerals

Prior to the development of our number system, there have been many other civilizations who have had their own unique way of handling mathematics and arithmetic. The one system that has survived to this day and is still in wide use is the Roman Numeral System.

The Roman Numeral System

A few major things to realize about the Roman Numeral System There is no zero It uses what we think of as letters (I, V, X, L, C, M) Placing a lower value to the left of a higher value subtracts Placing a lower value to the right of a higher value adds

I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 Notice the significance of 5 in this system (like 10 in our system)

See a pattern 1 = I 2 = II 3 = III 4 = IV 5 1 you add through three higher then for 4 you subtract

6 = VI 7 = VII 8 = VIII 9 = IX 10 1

LX = 60 50 + 10 XL = 40 50 10

Starting on the left you build the value MCMLXVI M = 1000 CM = 900 1000-100 LX = 60 50 + 10 VI = 6 5 + 1 1000 + 900 + 60 + 6 = 1966 you see these type of things on movie dates.

Lets go from our system to Roman

2121 2000 = MM 100 = C 20 =XX 1 = I MMCXXI

Least Common Multiple and Greatest Common Factor

When working with a group of two or more numbers, we sometimes have to find two specials numbers, the least common multiple and the greatest common factor.

Least Common Multiple

The Least Common Multiple is the smallest number that is a multiple of each number in the group. The least common multiple of 2 and 3 is 6 since it is the smallest number both 2 and 3 divide evenly

Finding the least common multiple. Factor each number Write down all the factors of the first number Add in the factors from the other numbers that are not in the LCM already Multiply all the numbers together

Ex. Find LCM for 30 and 45 30 = 2*3*5 45 = 3*3*5

LCM start with 30 and write down 2*3*5 Look at 45, it has 2-3s and a 5, the LCM has a 5 but only one 3 so we put in the

other 3 to get LCM = 2*3*5*3 = 90

Ex. Find LCM for 6. 8.15 6 = 2*3 8=2*2*2 15=3*5

LCM start with 6 and get 2*3 Go to 8 and put in 2 3s and get 2*3*2*2 Go to 15 and put in the 5 and get 2*3*2*2*5 LCM = 2*3*2*2*5 = 120

Greatest Common Factor

The greatest common factor is the largest number that is a factor of each number in the set of numbers. It is used in the reduction of fractions.

Finding the Greatest Common Factor Factor each number

Look at each factor in the first number and if it occurs in the other numbers (if it occurs in all numbers then include it in the GCF)

Ex. Find the GCF of 8 and 12 8 = 2*2*2 12 = 2*2*3

GCF look at first 2 in 8 it is also in 12 so get 2 so far 8 = 2*2*2 12 = 2*2*3 now go to the next 2 of 8 it is also in 12 so we get 2*2 so far 8 = 2*2*2 12 = 2*2*3 now go to the next 2 of 8 it is not in 12 so we have 2*2 = 4 as the GCF

Ex. Find the GCF of 60 and 200 60 = 2*2*3*5 200 = 2*2*2*5*5

look at things in common 60 = 2*2*3*5 200 = 2*2*2*5*5 so the GCF is 2*2*5 = 20

Reducing Fractions

Whenever we are dealing with numbers in the terms of fractions, we like to have them reduced to lowest terms.

The lowest terms of a fraction is the terms when the numerator and the denominator have no factors in common (relatively prime).

Ex.

3 is in lowest terms since 3 and 4 are relatively prime

4

8 is not in lowest terms since they have 4 as a common factor

20

To reduce fractions to lowest terms

Factor numerator and denominator

Cancel out factors in common

Ex.

8

Reduce to lowest terms

20

2 2 2 2becomes or

2 2 5 5

Ex.

44 Reduce to lowest terms

100

2 2 11 11 becomes or

2 2 5 5 25

Adding and Subtracting Fractions

There will be occasions where will be necessary to add or subtract numbers that are fractions. (I know we dont like fractions, but they are necessary).To add fractions they must have the same denominator (bottom). If they do not have the same denominator then we must convert each of them to a fraction with a common denominator.

Fractions with same denominators

Ex.

1 2 1+ 2 3+ = = , just add the numerators (tops) 4 4 4 4

Ex.

4 2 4 2 2 = = , just subtract numerators 15 15 15 15

Fractions with different denominators.

Ex.

1 1+ , different denominators. We must find a common multiple for the denominators to 2 3 use as a common denominator. The least common multiple of 2 and 3 is 6, so we use 6 as the common denominator. We convert each fraction to a new one with the denominator of 6.

1 1 3 3 3 = = multiply by 1 in the form of 2 2 3 6 3

1 1 2 2 2 = = multiply by in the form of 3 3 2 6 2

1 1 3 2 5+ = + = 2 3 6 6 6

http:necessary).To

Ex.

5 1 different denominators. The Least Common Multiple of 8 and 3 is 24 8 3

5 5 3 15 = = 8 8 3 24

1 1 8 8 = = 3 3 8 24

therefore

5 1 15 8 7 = = 8 3 24 24 24

Multiplying and Dividing Fractions

Multiplication of fractions is very simple, just multiply numerators and denominators

Ex.

1 3 13 3 = = 2 4 2 4 8

Ex.

2 3 6 1 = = 3 4 12 2

Division of fractions is not much harder but has one thing important to remember. You must invert the divisor and then multiply (Flip the last guy and multiply)

Ex.

2 3 2 4 2 4 8 = = = 3 4 3 3 3 3 9

Ex.

15 5 15 4 60 3 = = = 8 4 8 5 40 2

Mixed Numbers and Improper Fractions

There are two ways of expressing fractions representing numbers greater than one, mixed number and improper fractions.

Mixed numbers are expressed as a whole number part and a fractional part in the form B 1

A like 3 C 2

Improper Fractions are fractions whose numerator is larger than the denominator A

where A>B B

Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction

Multiply denominator by whole number part

Add numerator

Place over the denominator

Ex.

1

Convert 3 to an improper fraction

2

Multiply denominator by whole number part 3*2 = 6

Add numerator 6 + 1 = 7

7Place over the denominator

2

Ex.

3

Convert 5 to an improper fraction

4

5*4 = 20

20 + 3 = 23

23 so we get

4

Improper Fractions to Mixed Numbers

To convert improper fractions to mixed numbers, we have to remember the long division that we learned in elementary school division with remainders.

To convert from Improper Fractions to Mixed Numbers

Perform implied division with remainder

Write quotient as whole number part

Place remainder over divisor as fractional part

Ex.

Write 19

as a mixed number 9

1 2

9 19R so we get 9

12

Ex.

Write 23

as a mixed number 4

5 34 23R3 so we get 5

4

Introduction to Decimals

Numbers that cannot be represented as whole numbers are written as either fractions or in decimal notation. We are familiar with the concept of decimal notation from numerous examples in our lives, namely the use of money ($3.12 is decimal notation for 3 dollars and 12 cents)

We can think of decimal notation as another way of writing certain special types of fractions (those with multiples of ten in the denominator)

10

3 Three tenths 0.3 Note: 1 zero in the denominator and 1 decimal place

100

3 Three hundredths 0.03 Note: 2 zeroes in the denominator and 2 decimal places

1000

239 Two hundred thirty-nine thousandths

0.239 Note: 3 zeroes in the denominator and 3 decimal places

We should be able to note that there are exactly three parts to a decimal number.

3.12

whole number part decimal point decimal part

Writing decimals numbers in words

0.03 is read as 3 hundredths since the 3 is in the second decimal place (1/100)

0.6481 is read as six thousand four hundred eighty-one ten-thousandths since the 1 is in the fourth decimal place (1/10000)

Writing decimal numbers in standard form

Five and thirty-eight hundredths hundredths implies a total of 2 decimal places to be filled by the 38 so we get 5.38

Nineteen and four thousandths thousandths implies a total of 3 decimal places to be filled by 4 so we add two leading zeroes to make 004 and get 19.004

Rounding decimals

Sometimes we are called upon to limit the number of decimal places that can be used in a specific application (it makes no sense to take money out to 3 places). This process is known as Rounding.

Rounding rules If the number to the right of the given place value is less than 5, drop that number and all numbers to the right of it.

If the number to the right of the given place value is 5 or greater, increase the number in the given place value by one and drop all numbers to the right of it

Round 26.3799 to the nearest hundredth Look at 26.3799, 7 is in the hundredths (second) place and 9>5 so increase 7 to 8 and drop the 99 and get 26.38

Round 42.0237412 to the nearest hundred thousandth Look at 42.0237412, 4 is in the hundred thousandth (fifth) place and 1 < 5 so drop the 12 and get 42.02374

Addition of decimals

The addition of decimal numbers is almost identical to the addition of whole numbers. The only difference is that we need to remember to line up the vertical columns with respect to the decimal point.

Add 0.237 + 4.9 + 27.32

27.32 4.9 0.237

+ Remember to line up on the decimal points

27.320 04.900 00.237

+ Rewrite with zeroes added in appropriate place values to make things

line up properly

32.457 27.320 04.900 00.237

+

7 + 0 + 0 = 7 3 + 0 + 2 = 5

2 + 9 + 3 = 14 carry 1 0 + 4 + 7 + (1) = 12 carry 1

0 + 0 + 2 + (1) = 3

Subtraction of decimals

Subtraction of decimals is almost identical to subtraction of whole numbers. The only difference is to remember to line up the columns on the decimal point. (all rules of borrowing in subtraction apply)

Subtract 6.93 3.7

3.7 6.93

Add necessary zero to line up columns

3.70 6.93

3 0 = 3 9 7 = 2 6 3 = 3

3.23 3.70 6.93

Subtract 39.047 7.96

7.960 39.047

7 0 = 7

7 7.960

39.047

We need to borrow from the 9 (changed to 8) to change the 0 to 10 so that we can borrow from the 10(changed to 9) to change the 4 to 14

87 031. 60 7. 9 47 39 .0 198

// So the result is 31.087

Multiplication of Decimal Numbers

Decimal numbers are multiplied in the same way as whole numbers, with special consideration given to the number of decimal places in each of the factors. (number of decimal places in first factor + number of decimal places in second factor = number of decimal places in product)

Multiply 21.14 0.36

21.4 1 decimal place 0.36 2 decimal places

1284 642 1 + 2 = 3 7.704 3 decimal places

Multiply 0.0370.08

0.037 has 3 decimal places 0.08 has 2 decimal places 37 8 = 296 to make this number have 5 decimal places, we need to add 2

addition zeroes 0.00297 (the idea of keeping track of decimal places was necessary in the days when we used slide rules to perform our multiplications. The slide rule would do the whole number multiplication for us (3 8 = 297) but you had to put the decimal places in for yourself)

Multiplication by multiples of ten

To multiply by multiples of ten, move the decimal point to the left the same number of places as there are zeroes in the

multiple of ten factor.

3.82 10 = 38.2 1 zero and 1 move to the left 3.82 100 = 382. 2 zeroes and 2 moves to the left 3.82 1000 = 3820. 3 zeroes and 3 moves to the left (needed to add a 0

at the right end to make the move)

Division of Decimal Numbers

To divide decimal numbers, move the decimal point in the divisor to the right enough place to make it a whole number. Also move the decimal point in the dividend an equal number of places.

(remember to keep the decimal point in the quotient directly above the decimal point in the dividend)

Divide 3.25 15.275

We need to change 3.25 to 325 by moving the decimal point 2 places to the right Therefore, we need to change 15.275 to 1527.5 by moving the decimal point 2

places to the right. So the problem becomes 325 1527.5

4.7 5.1527325

1300 2275 note: the decimal point in 4.7 is above the decimal point in1527.5

2275 0

Not all divisions of decimal numbers will come out even as in the above example. We generally round the quotient in a decimal division instead of carrying a remainder.

Divide 0.3 0.56 and round to two decimal places

First convert the problem to 3 5.6 , to be able to round to two decimal places we

must have 3 decimal places in the quotient so we will arrange to have 3 decimal

places in the dividend 3 5.600

1.866 3 5.600 3

2 6 We will round 1.866 to 1.87 2 4

20 18

20 18

Dividing by multiples of ten

To divide by multiples of ten, move the decimal point to the left the same number of places as there are zeroes in the multiple of ten divisor (placing leading zeroes as necessary)

34.65 10 = 3.465 1 zero in 10 so move 1 place left 34.65 100 = 0.3465 2 zeroes in 100 so move 2 places left 34.65 1000 = 0.03465 3 zeroes in 1000 so move 3 places left (needed to add a leading zero for the move)

Converting between Decimals and Fractions

Fractions to Decimals

To convert from a fraction to a decimal it is as simple as performing the implied division as we learned in elementary school.

Terminating Decimals (division ends) Ex.

2.55 becomes 2.5 since 2 5.0

2

.25 1 becomes .25 since 4 1.00

4

Non-terminating Decimals (repeating decimals) Ex.

.3333 1 becomes .3333 since 1 3.0000 repeating 3s forever

3

.1414 14 becomes .1414 since 99 14.00 repeating 14s forever

99

Decimals to Fractions

Remember that decimals are actually fractions with the denominator as an appropriate power of ten (the number of zeroes after the one is equal to the number of places to the right of the decimal point)

Ex. 5 1

.5 has 1 place to right so it is reduced to 10 2

125 1.125 has 3 places right so it is reduced to

1000 8

34 2 3.4 has 1 place right so it becomes reduced to 3

10 5

Scientific Notation

Occasionally, we need to deal with very large or very small numbers. It is convenient to use a system called scientific notation to represent these numbers. Scientific notation is based on powers of ten to represent these numbers.

Large numbers

10 1 = 10 10 2 = 100

3 notice the exponent indicates the number of zeroes 10 = 1000 10 4 = 10,000

We can express large numbers as a number between 1 and 10 multiplied by the appropriate power of 10

289 = 2.89 102 notice the decimal point moved 2 units to the left (same as exponent)

36782 = 3.6782 104 4 place move and exponent of 4

Small Numbers

10-1 = 0.1

10-2 = 0.01

10-3 = 0.001 notice the exponent indicates the number of places right of decimal point

10-4 = 0.0001

We can express small numbers as a number between 1 and 10 multiplied by the appropriate power of 10.

0.234 = 2.34 10 1 we moved the decimal point 1 unit to the right

0.000123 = 1.23 10 4 we moved the decimal point 4 units right

Introduction to Ratios

In real life, numbers are usually quantities of objects and have units associated with them. (6 balls, 12 feet and 9 cars)

A ratio is a comparison between two numbers that have the same units.

Ratios can be written in three different ways 3 feet 3 As a fraction = , ratios do not have units 4 feet 4

As two numbers separated by a colon 3 feet : 4 feet, 3 : 4 As two numbers separated by the word to 3 feet to 4 feet, 3 to 4

Compare two boards, one 6 feet long to another of 8 feet long. 6 feet 6 3 = = , this means that the shorter board is the length of the longer 8 feet 8 4

board

Introduction to Rates

A rate is the comparison of two quantities that have different units.

Rates are written as fractions

A cyclist rode 144 miles in 10 hours. We can write a distance to time ratio for the trip 144 miles 72 miles = , always simplify the fraction, whenever possible 10 hours 5hours

Unit rates

Whenever a rate has 1 as the denominator, it is referred to as a unit rate.

$6.25 $3.25 2 pounds of sirloin sell for $6.25, = , or $3.25 per pound

2lbs 1lb

360 miles 60 miles A car travels 360 miles in 6 hours, = , 60 miles per hour

6hours 1hour

Introduction to Proportions

A proportion is an expression of the equality of two ratios or rates.

50 miles 25 miles = is a proportion 4gals 2gals

A proportion is true if the two fractions are equal (when written in lowest terms). The best way to check if a proportion is true is to check the equality of the cross products

a c.i.e. given = is true if ad = bc

b d

2 8Is = a true proportion?

3 12 2*12 = 24 3*8 = 24 24 = 24 so it is a true proportion

A proportion is false if the cross products are not equal.

4 8Is = a true proportion?

5 9 4*9 = 36 5*8 = 40 therefore a false proportion

Solving proportion problems

Sometimes, we do not know one of the values of a proportion. We use the above mentioned property of cross products to solve for the missing value

9 3Solve =

6 n By cross multiplying we get 9n = 6(3)

9n = 18 9n 9 = 18 9

n = 2 9 3

thus = is a true proportion 6 2

Introduction to Percents

Percents, like fractions and decimals, are ways of writing numbers that are not necessarily whole numbers.

Percent means parts of 100, so they can always be written as fraction with 100 in the denominator.

Writing percents as a fraction 1

To write percents as a fraction, multiply the percent by 100

1 13 1 13 13% = 13 = =

100 1 100 100 1 120 1 120 6 1

120% = 120 = = = = 1 100 1 100 100 5 5

2 2 1 50 1 50 1

16 % = 16 = = =

3 3 100 3 100 300 6

Sometimes it helps to change mixed numbers to improper fractions for the calculation

Writing percents as decimals

To write a percent as a decimal, multiply the percent by 0.01 (move decimal point two places to the right)

5% = 5 * 0.001 = 0.05 215% = 215 * 0.001 = 2.15

Writing fractions or decimals as percents

To write fractions or decimals as percents, multiply the fraction or percent by 100%

Change3/8 to a percent 3 3 100 300 100% = % = % = 37.5% 8 8 1 8

Change 0.48 as a percent 0.48 * 100% = 48%

Table of Common Fractions and Their Percentage Equivalents

1/2 = 50%

1/3 = 33 1/3%

2/3 = 66 2/3%

1/4 = 25% 3/4 = 75%

1/5 = 20% 2/5 = 40%

3/5 = 60% 4/5 = 80%

1/6 = 16 2/3%

5/6 = 83 1/3%

1/7 = 14 2/7%

2/7 = 28 4/7%

3/7 = 42 6/7%

4/7 = 57 1/7%

5/7 = 71 3/7%

6/7 = 85 5/7%

1/8 = 12 1/2%

3/8 = 37 1/2%

5/8 = 62 1/2%

7/8 = 87 1/2%

1/9 = 11 1/9%

2/9 = 22 2/9%

4/9 = 44 4/9%

5/9 = 55 5/9%

7/9 = 77 7/9%

8/9 = 88 8/9%

1/10 = 10% 3/10 = 30%

7/10 = 70% 9/10 = 90%

1/12 = 8 1/3%

Table of Common Fractions and Their Decimal Equivalents or Approximations

1/2 = 0.5 1/3 = 0.3333...

2/3 = 0.6666... 1/4 = 0.25

3/4 = 0.75 1/5 = 0.2

2/5 = 0.4 3/5 = 0.6

4/5 = 0.8 1/6 = 0.1666...

5/6 = 0.8333... 1/7 = 0.142857142857...

2/7 = 0.285714285714... 3/7 = 0.428571428571...

4/7 = 0.571428571428... 5/7 = 0.714285714285...

6/7 = 0.8571428571428... 1/8 = 0.125

3/8 = 0.375 5/8 = 0.625

7/8 = 0.875 1/9 = 0.111...

2/9 = 0.222... 4/9 = 0.444...

5/9 = 0.555... 7/9 = 0.777...

8/9 = 0.888... 1/10 = 0.1

3/10 = 0.3 7/10 = 0.7

9/10 = 0.9 1/12 = 0.08333...

Percent Equations

There are many ways to solve problems involving percents. All of them require that you identify three parts of the problem (percent, base and amount), but most of require that you also learn rules for when you multiply and when you divide. This is fairly confusing.

The best way to solve the basic percent problem is to use the proportion, percent amount = and the cross multiplication, percent * base= 100 * amount.

100 base

4% of 85,000 is what number? First, we must determine where the above numbers go in the proportion. As a

amount general rule, you can look at the ratio by looking at the wording of the

base problem. The number associated with the word of generally as the base and the

amount is number associated with the word is generally is the amount. So, = .

base of 4 = percent is = what number (the variable (n)) of = 85,000

4 n = , by cross multiplying we get 100 85,000 4(85,000) = 100n 340000 = 100n 3400 = n

What percent of 40 is 30? Is = 30 Of = 40 Percent = what (n)

n 30 = 100 40

40n = 3000 n = 75%

18% of what is 900? Percent = 18 Is = 900 Of = what (n) 18 900 =

100 n 18n = 90000 n = 5000

Applications of Percents

A certain charitable organization spent $2940 for administrative costs. This is 12% of the total amount of the monies they collected. How much did they collect in total? We are really asking $2940 is 12% of what number?

12 2940 = 100 n 12n = 294000 n =24,500 so, they collected a total of $24,500

The fire department reports that 24 false alarms were sent in out of a total of 200 alarms. What percent of all the alarms are false? We are really asking 24 is what percent f 200?

n 24 = 100 200

200n = 2400 n = 12% so, 12% were false alarms

An antiques dealer claims that 86% of her sales are for items that cost less than $1000. If she sells 250 items in a given month, how many will sell for less than $1000? We are really asking what number is 86% of 250? (notice $1000 is not used

anywhere) 86 n =

100 250 100n = 21500 n = 215 so, 215 sold for less than $1000

Negative Numbers

There are special numbers associated with the positive numbers that are called the additive inverses. These are numbers that when added to a positive number will give 0 as the result.

Ex. 4 + something = 0 something = -4, therefore 4 + -4 = 0

6.6 + -6.6 = 0 1024 + -1024 = 0

These additive inverses of positive numbers are called negative numbers. Many of us are familiar with negative numbers when we look at our checking accounts (the US Government).

Adding and Subtracting Signed Numbers

There are basic rules used in the addition and subtraction of signed (+ or -)

Addition of signed numbers

There are two situations to consider in the addition of two signed numbers (or the signs the same or different)

Signs the same

If the signs of the two numbers are the same, just add the numbers and apply the common sign

Ex.

4 + 5 = 9 Both are positive so 9 is positive

(-4) + (-5) = (-9) Both are negative so -9 is negative

Signs different

If the signs of the two numbers are different then subtract the two numbers and use the sign of the number with the larger absolute value

Ex.

4 + (-5) 5 4 = 1 and since 5 > 4 we use the from -5 to get -1

-41 + 51 51 41 = 10 and 51 > 41 so we get +10 (10)

Subtraction of Signed Numbers

The best way to subtract signed numbers is to not do it. You can always convert a subtraction into an addition and work from there.

Ex.

4 5 convert to 4 + (-5), the conversion to to addition requires a sign change in the number after the subtraction sign

4 + (-5) = -1 by addition rules above

Ex.

-4 5 convert to -4 + (-5) = -9

Ex. 4 (-5) convert to -4 + 5 (note -5 changed to 5) so -4 + 5 = 1 by addition rules

When it comes to larger, more complicated problems, remember you can only add or subtract two numbers at a time, so only work with two at a time

Ex.

4 + (-5) 6 + (-7) 9 pretty long, but take two at a time

4 + (-5) = -1 so we get

-1 6 + (-7) 9

-1 6 = -1 + (-6) = -7 so we get

-7 + (-7) 9

-7 + (-7) = -14 so we now get

-14 9 = -14 + (-9) = -23

Multiplying and Dividing Signed Numbers

There are exactly two rules that must be followed when multiplying or dividing two signed numbers

If the signs agree, the answer is positive

If the signs disagree, the answer is negative

Remember, the operations of multiplication and division can only be done with 2 numbers at a time complicated problems must be taken two numbers at a time.

Ex.

4*(-5) = -20 opposite signs

(-4)*(-5) = 20 same signs

20 = 5 opposite signs 4

20 = 4 same signs 5

Ex.

(-4)(-5)(2)(-6) (-4)(-5) = 20

20(2)(-6) (20)(2) = 40

40(-6)

-240

Variables

Throughout our study of algebra, we hear about these things called variables.

What is a variable?

Variables are just symbols to represent values that are currently unknown. They stand in the place of numbers.

Ex. In the expression 3x, x represents a number and is called a variable

In the expression 2x -4y, x and y both represent different numbers and are called variables

Throughout algebra, and actually real life, we encounter unknown values that can be expressed as variables. If you go to the grocery store and find that 3 cans of cream corn cost $1.29, the price of one can is unknown and can be represented by a variable (maybe p for price)

Evaluating Algebraic Expressions

Algebraic expressions will have a value dependent upon the number that is substituted into the variable.

Example: Evaluate 4x + 3 if x = 1 4(1) + 3 = 7 + 3 = 10

Evaluate 4x + 3 if x = -2 4(-2) + 3 = -8 + 3 = -5

Combining Like Terms in Algebraic Expressions

Two or more terms in an algebraic expression that have the same variable part are called like (similar) terms. (Think of apples and oranges. If an expression is made up of 3 apples and 2 oranges, they cannot be combined since apples oranges)

If an algebraic expression contains 2 or more like terms, they can be combines to form a single term.

Examples: 4x + 7x = (4 + 7) x = 11x 13u 7u + 2u = (13 7 + 2) u = 8u 6x + 3x + 7y = (6 + 3) x + 7y = 9x + 7y

Not like terms

Properties of Equations

The major thing to remember when working with equations, is when you make a change to one side of the equation you must make an identical change to the other side.

Rule: You can add or subtract a value to both sides of an equation without changing the solution to the equation.

Examples; X + 3 = 7, if I add 3 to the left, I must add 3 to the right X + 3 + 3 = 7 + 3 X + 6 = 10

Y + 6 = 10, if I subtract 6 from the left, I must subtract 6 from the right Y + 6 6 = 10 6 Y = 4

Rule: You can multiply both sides of the equation by a constant or divide a constant into both sides of the equation without changing the solution to the equation.

Examples:

-4c = 20 I can divide both sides by 4 -4c -4 = 20 -4

c = 5

2 5 x = I can multiply both sides by 3/2

3 9 3 2 3 5 x = 2/3 * 3/2 =1 2 3 2 9

15 5 x = =

18 6

You can also combine both of the above rules in the same problem to simplify.

3x + 2 = 8 Subtract 2 from both sides 3x + 2 2 = 8 2

3x = 6 Divide both sides by 3 3x 3 = 6 2

x = 2

1 x 2 = 6 Add 2 to both sides

2 1

x 2 + 2 = 6 + 2 2

1 x = 8 Multiply both sides by 2

2 1

x 2 = 8 2 2

x = 16

Algebraic Equations

When we connect two algebraic expressions using = , we call the combination an equation.

There are three parts to an equation

X + 3 = 7

Left side equal sign right side

An equation can be true or false, dependent upon the number substituted into the variable.

Examples X + 3 = 7 is true if X = 4 X + 3 = 7 is false if X = 1

Some equations are true for any value of the variable. These types of equations are called identities.

Example: 2(3x + 7) = 6x + 14 is an identity

x = 5 2(3(5) + 7) = 6(5) + 14 44 = 44

x = 1 2(3(1) + 7) = 6(1) + 14 20 = 20

Most equations are not identities. They are true for only certain values of the variable. These are called conditional equations.

Example: 5x 2 = 3 is true if x = 1 5x 2 = 3 is false if x = any other number 5x 2 = 3 is a conditional equation

The process of determining the values that make a conditional equation true is called solving the equation.

Solving Linear Equations

Linear Equation An equation where the only exponent involved with the variable is 1 is called a linear equation.

Examples: 3x + 2 = 8 2(x + 4) x + 7 = 6 are both linear equations

The best way to look at the procedures for solving a linear equation is to undo the operations that have been done to the variable.

Example:

Equation Done to x How undone 3x + 2 = 8 2 added to 3x Subtract 2

3x + 2 2 = 8 2 3x = 6 Multiplied by 3 Divide by 3

3x 3 = 6 2 x =2

We also need to be able to get the variables on one side and the constants on the other sides.

Example:

3x + 2 = x 6 Move x to left by subtracting x from both sides

3x + 2 x = x 6 + x 2x + 2 = 6 Subtract 2 from both sides

2x + 2 2 = 6 2 2x = 4 Divide both sides by 2

2x 2 = 4 2 x = 2 Done

Procedures for Solving Linear Equations

1. Perform any implied operations (multiply through parenthesis, etc.) 2. Combine like terms on each side 3. Perform necessary operations to separate constants and variables 4. Combine like terms as necessary 5. Multiply or divide to make the numerical coefficient 1

Literal Equations

Literal equations are also called formulas. They form a relationship between two or more variable, such as the formula to find the area of a circle from the radius A = r 2 . Occasionally, we will need to solve for the variable that is not explicitly solve for in the equation, like trying to find r from the formula above.

To solve for a variable in a formula, treat the other variables as constants and solve as you normally would any other equation.

Ex. Solve A = 2L + 2W (area of a rectangle) for L

A = 2L + 2W subtract 2W from each side

A 2W = 2L divide both sides by 2

(A-2W)/2 = L given A and W can find L

Solve C = 2 r for r (circumference of a circle)

C = 2 r divide both sides by 2 (remember is just a number)

C/2 = r can find r given C

Linear Inequalities

The rules for solving linear inequalities are the same as those for linear equalities except for one thing, if you multiply or divide the inequality by a negative number the inequality sign switches.

The solutions to linear inequalities are intervals of numbers, not individual numbers like in equalities.

Interval Notation

There are 2 symbols used to show the endpoints of intervals

( - endpoint of the interval not included

[ - endpoint of interval included

Ex.

(0,2) all numbers between 0 and 2 but not including 0 or 2

[0,2) all numbers between 0 and 2 including 0 but not 2

[0,2] all numbers between 0 and 2 including 0 or 2

(- , 2] all numbers less than or equal to 2

(2, ) all numbers greater than 2 but not 2

Solving Inequalities

Ex.

Solve 3x + 2 > 0 subtract 2 from each side

3x > -2 divide by 3

x > -2/3 (-2/3, )

Solve -4x 8 < 12 add 8 to each side

-4x < 20 divide by -4(remember to switch sign)

x > -5 (-5, )

Solve 3x + 4 < 2x 1 put xs on one side and constants on the other

3x + 4 < 2x 1 subtract 4 from each side

3x < 2x 5 subtract 2x from each side

x < -5 (- , -5)

Multiplication using Exponents

Exponents are a shortcut form for repeated factors in a multiplication.

Examples; 34 = 3*3*3*3 = 81 (-2)5 = (-2)*(-2)*(-2)*(-2)*(-2) = -32

2 2 2 2 8( )3 = = 3 3 3 3 27

n times

n Parts of the exponential number x

xxxxxn =

Base Exponent

There are a few rules governing the use of exponential numbers

Rule 1 Multiplication Property of like terms am * an = am+n

34 * 37 = 34+7 = 311

Rule 2 Power to a Power Property (am)n = amn

(42)6 = 42*6 = 412

Rule 3 Power of a Product Property

nbn(ab)n = a

(5x)2 = 52x2 = 25x2

(-x)5 = (-1)5x5 = -x5

(-x)6 = (-1)6x6 = x6

Further Examples:

(2x3)4 = 24x3*4 = 24x12 = 16x12

3)3(-3xy2)2(2x3y2 2*223 3*3 3*3 (-3)2x y x y

9x2y48x9y9

72 x11y13

Division with Exponents

Look at

naIn order to evaluate

m we need to develop a few rules that will lead us to a some

a interesting properties of exponents.

note:5 3 = 2

na nm= aTherefore we have a rule ma

Examples: 65 64 2 4

= 5 = 5 5 64 0= 6 64

64 60also

4 = 1 therefore = 1

6

0From here, we get a = 1 Zero Power Rule

54 46 2= 5 = 5 56

54 5*5*5*5 1 1 1also = = = therefore 52 =

6 2 25 5*5*5*5*5*5 5*5 5 5

n = 1 leading us to the a na Negative Power Rule

)4

3( 2 = )

4

3)(

4

3( =

4

3 2

2

= 16

9

:

giving us n

b

a )( =

n

n

b

a Power of a Quotient Rule

Further Examples:

4 4*5 20 x x x( )5 = ( ) =

3 3*5 15 y y y

2 4 3 2 12 2 12 69x ( y ) 9x y 9 x y 1 6 y= = = 1 y = 2 3 2 4 6 4 6 2 2(3x y ) 9x y 9 x y x x

note: the variable ends up on the side of the fraction line as the largest power

6 3 2 2 3*2 2*2 6 4 x(x y ) = x y = x y =

4 y note: the negative power causes the variable to change sides of the fraction line

Polynomials

Polynomials are a special type of algebraic expression where the exponents are positive integers.

Ex.

x2 + 2x -1 is a polynomial since exponents are 2 and 1

x3/2 is not a polynomial since exponent is 3/2

x is not a polynomial (radical variables have fractional exponents)

3x is a polynomial since exponent is 1

Evaluating Polynomials

Polynomials will have a value dependent upon the number that is substituted into the variable.

Example: Evaluate 4x + 3 if x = 1

4(1) + 3 7 + 3 10

Evaluate 4x + 3 if x = -2 4(-2) + 3 -8 + 3 = -5

Evaluate y3 + 3y 5 if y = -3 (-3)3 + 3(-3) 5 -27 - 9 5

-40

Evaluate y3 + 3y 5 if y = 0 (0)3 + 3(0) 5 0 + 0 5 -5

Evaluate 3u2 + 2uv 12 if u = -1 and v = 5 3(-1)2 + 2(-1)(5) 12 3(1) 10 12 -19

Adding Polynomials

Adding polynomials is basically the same as the addition of numbers with units.

You can only add terms that are similar.

Ex.

2 apples + 3 oranges + 5 apples = (2 + 5) apples + 3 oranges = 7 apples + 3 oranges (you cant add apples and oranges)

Ex.

2x + 3x = (2 + 3)x = 5x

Ex.

5y + 3x + 8y + 4x = (5 + 8)y + (3 + 4)x = 13y + 7x

Ex.

2x2 +3x2 + 4x3 = 5x2 + 4x3 (note:x2 and x3 are not similar terms)

Subtracting Polynomials

The subtraction of polynomials is basically the same as the addition of polynomials, with one exception (you have to deal with the subtraction sign). We accomplish this by remembering how to subtract signed numbers change the subtraction to addition and change the sign of everything after the minus sign.

Ex.

(3x2 + 2x + 4) (-2x2 + 3x + 1) change to

(3x2 + 2x + 4) + (2x2 3x 1) combine like terms

(3x2 +2x2) + (2x 3x) + (4 1) perform operations

5x2 -1x + 3

Ex.

(3xy 4y + 3xz) (3y -3xy +xz) change to

(3xy - 4y + 3xz) + (-3y + 3xy xz) combine like terms

(3xy +3xy) + (-4y +3y) + (3xz xz) perform operations

6xy y + 2xz

Multiplying Polynomials

Multiplying polynomials is mostly a trial in keeping things lined up and not missing any parts of the problem.

Technique to multiply polynomials Break into several simple problems using distribution Multiply each part Combine like terms

Ex. (3x2 + 2)(3x 1) use distribution of first term 3x2(3x -1) + 2(3x 1) distribute again 3x2(3x) -3x2(1) + 2(3x) 2(1) Multiply terms 9x3 3x2 + 6x -2

Ex. (6x +2y + 1)(2x -3y 2) 6x(2x -3y 2) + 2y(2x 3y 2) + 1(2x -3y 2) 6x(2x) 6x(3y) -6x(2) +2y(2x) 2y(3y) -2y(2) +1(2x)-1(3y)-1(2) 12x2 -18xy-12x + 4xy 6y2 -4y + 2x 3y -2 12x2 -18xy +4xy -12x +2x -4y -3y -6y2 -2 12x2 -14xy -10xy -7y - 6y2 2

FOIL Method If you are multiplying a binomial by a binomial you can use a variation of the above technique call First Outside Inside Last (FOIL)

(ax + b)(cx + d)

First Outside Inside Last

(ax)(cx) = acx2

ax(d) b(cx) bd acx2 +(ad + bc)x + bd

Ex. (x + 2)(x + 3) First Outside Inside Last

x(x) x(3) 2(x) 2(3) x2 +3x +2x + 6 x2 + 5x + 6

Special Products

There are a few special types of products that are handy to keep in your back pocket for use in a hurry. These are not necessary to memorize, but they can help you perform multiplication of polynomials rapidly, in the cases where they apply.

Special Product #1

2 2 2(a + b) = a + 2ab + b Example

2 2 2 2(6 + y) = 6 + 2(6)( y) + y = 36 +12 y + y

Special Product #2

2 2 2(a b) = a 2ab + b Example:

2 2 2 2(6 y) = 6 2(6)( y) + y = 36 12 y + y

Special Product #3

(a + b)( a b) = a 2 b 2 Example:

(3 + c)(3 c) = 32 c 2

Further Examples using the Special Product Rules

2 2 2 2(8 + 2y) = 8 + 2(8)(2y) + (2y) = 64 + 32 y + 4y

3 2 1 2 3 2 2 3 2 1 1 2 6 4 3 2 1(4m n ) = (4m n ) 2(4m n )( ) + ( ) = 16 m n 4m n +2 2 2 4

2 22 3 2 3 2 3 4 6(3x 2 y )(3x + 2y ) = (3x ) (2 y ) = 9x 4 y

Dividing Polynomials

There are two types of division problems for polynomials, those that can be done as fractions and those that require long division. If the divisor is a single value then the division can be broken into a group of simple fractions, otherwise you must use long division.

Division using fractions

Ex.

18 2x 4 x = 18 2x 4x = 9x 2

2x 2x 2x

Long Division

Ex.

x2 divided by x gives us x

x

121 2 +++ xxx

121 2 +++ xxx (x 2 + x) multiply x(x+1) and subtract

x +1

x divided by x gives 1

1 121 2

+ +++

x xxx

(x 2 + x) x +1 Multiply 1(x + 1) and subtract

(x +1) 0

Therefore, we get (x 2 + 2x + 1) (x + 1) = x + 1

Common Factors

We can expand the idea of the Greatest Common Factor, that is studied in earlier classes into the world of algebra. We will be looking for the common factors in algebraic expressions.

Look at 30x2 and 42x3. We can factor them individually. 30x2 = 2 * 3 * 5 * x * x 42x3 = 2 * 3 * 7 * x * x * x

Each term has a 2, a 3, and 2-xs so they have 2 * 3 * x * x = 6x2 in common 30x2 = 6x2(5) 42x3 = 6x2(7x)

Example: Find the common factors for 15x3y3, 6x2y3 and 9xy4

First look at the coefficients 15, 6, and 9 The common factor of 15, 6, and 9 is 3

Look at the x variables The exponents on x are 3, 2, and 1 choose the lowest number, x1 = x is the common x term

Look at the y variable The exponents of y are3, 3, and 4 choose the lowest number, x3 is the common term.

Therefore the common term is 3xy3

15x3y3 = 3xy3 (5x2) 6x2y3 = 3xy3 (2x) 9xy4= 3xy3 (3y3)

Factoring polynomials by common factors

A polynomial is factored when it is expressed as a product of 2 or more polynomials.

The distributive property of integers and algebra, P(Q + R) = PQ + PR is the greatest aide in factoring. For the purposes of factoring we generally write the property in reverse order PQ + PR = P(Q + R). In this way we can see that P and (Q +R) are the factors of PQ + PR.

Example: Factor 5x + 30 by common factors Since 5x = 5 * x and 30 = 5 * 6, 5 is the common factor 5x + 30 = (5 * x) + (5 * 30) 5x + 30 = 5(x + 30) by reverse distributive law.

Procedure to factor by common factors.

1. Find the common factor in all the terms 2. Write each term as a product using the common factor found in step 1 3. Use reverse distributive law to factor out the common factor

Example: Factor 5y3 + 25y

The common factor in the terms is 5y 5y3 + 25y = 5y(y2) + 5y(5)

so 5y3 + 25y = 5y(y2 + 5)

Example: Factor 18h5 + 12h4 21h3

The common factor is 3h3

18h5 = 3h3(6h2) 12h4 = 3h3(4h) 21h3 = 3h3(7)

so 18h5 + 12h4 21h3 = 3h3(6h2) + 3h3(4h) - 21h3 - 3h3(7) 18h5 + 12h4 21h3 = 3h3(6h2 + 4h 7)

Example: Factor 4a7b5c + 10a3b8 16a4b6 + 18a5b7

The common factor is 2a3b5, note c is not in all terms 4a7b5c = 2a3b5(2a4c) 10a3b8 = 2a3b5(5b3) 16a4b6 = 2a3b5(8ab) 18a5b7 = 2a3b5 (9a2b2)

so 4a7b5c + 10a3b8 16a4b6 + 18a5b7 = 2a3b5(2a4c) + 2a3b5(5b3) - 2a3b5(8ab) + 2a3b5 (9a2b2)

4a7b5c + 10a3b8 16a4b6 + 18a5b7 = 2a3b5(2a4c + 5b3 - 8ab + 9a2b2)

Factoring by Grouping

Certain polynomials, though they have no common factors, can be still be factored by using common factors. We have to be able to re-group the terms so that they are in groups that have common terms.

Look at 3xm + 3ym 2x 2y 2 terms have x and 2 terms have y (3xm 2x) + (3ym 2y) Factor x out of the first part and y out of the last part x(3m 2) + y(3m 2) note: 3m 2 is now in common (x + y)(3m 2)

Example: Factor 3a + 3b ma mb (3a + 3b) + (-ma mb) Factor 3 out of first part and m out of second part 3(a + b) m(a + b) note: factoring m out of mb leaves +b (3 m)(a + b)

Solving Equations by Factoring

We can use factoring to solve algebraic equations. We will use the property of multiplication of two numbers equaling zero, (if a*b = 0 then either a = 0 or b = 0)

Solve x2 + 5x = 0 By factoring we get x(x + 5) = 0 therefore x = 0 or x + 5 = 0 which implies x = 0 or x = -5

Solve 2x2 6x = 0 By factoring we get 2x(x 3) = 0 therefore 2x = 0 or x 3 = 0 which gives us x = 0 or x = 3

Solve x2 +5x + 6 = 0 By factoring we get (x + 3)(x + 2) = 0 therefore x + 3 = 0 or x + 2 = 0 which leads to x = -3 or x = -2

Factor the difference of two squares

We know from our multiplication rules that (a + b)(a b) = a2 b2. Applying this rule backwards we get a factoring rule for the difference of two squares,

a2 b2 = (a + b)(a b)

If we can identify the problem as being the difference of two squares, we can apply above rule.

Example: Factor x2 4

x2 22 a = x and b = 2 (x + 2)(x 2)

Example: Factor 4t2 9s2

(2t)2 (3s)2 a = 2t and b = 3s (2t + 3s)(2t 3s)

Example: Factor u4 16

(u2)2 - 42 a = u2 and b = 4 (u2 + 4)(u2 4) (u2 + 4)(u2 22) a = u and b = 2 (u2 + 4)(u + 2)(u 2)

Factoring the sum or difference of cubes

If a polynomial is made up of the sum of difference of two perfect cubes, we have special rules to handle the factoring (these are not easy to memorize, I can never remember them without looking).

a3 + b3 = (a + b)(a2 ab + b2) a3 - b3 = (a - b)(a2 + ab + b2)

Example: Factor x3 + 8 note: 8 = 23

Therefore, we get x3 + 23 a = x and b = 2 (x + 2)(x2 2x +22) (x + 2)(x2 2x +4)

Example: Factor 64u3 27v6 note: 64u3 = (4u)3 and 27v6 = (3v2)3

(4u)3 - (3v2)3 a = 4u and b = 3v2

(4u 3v2)((4u)2 + (4u)(3v2) + (3v2)2) (4u 3v2)(16u2 + 12uv2 +3v4)

Introduction to Rational Expressions

A rational expression is a fraction (p/q) where p and q are polynomials.

3 2y 7c +1Ex. , , are all rational expressions

x y +1 c 2 1

Evaluation of Rational Expressions As with polynomials the value of a rational expression is dependent upon the

value chosen for the variable.

x + 2Ex. Evaluate for x = -1, x = 2, and x = 1

x 1

1+ 2 1 1 x = -1 gives us = =

11 2 2

2 + 2 4x = 2 gives us = = 4

2 1 1

1+ 2 3x = 1 gives us = which implies there is no solution

11 0

It is important to realize the denominator (bottom) of a fraction can not equal to zero. Zero in the denominator of a fraction causes the fraction to be undefined.

x 5Ex. For what value of x is undefined?

3x +1

For a rational expression to be undefined, the denominator must be equal 1

to zero therefore we set 3x + 1 = 0 and get x = - . Thus we cannot 3

1 x 5substitute - into the fraction

3 3x +1

Equivalent Rational Expressions

If two rational expressions can be reduced to the same rational expression, then the two rational expressions are called equivalent rational expressions

6x 2Examples: and are equivalent fractions.

9x 3

6 y 2 yand are not equivalent

2y + 6 y + 3

2y is in reduced form

y + 3

3y 2y y + 3 y + 3

Equivalent fractions in signed number systems p p p= = q q q

p p p p= = = q q q q

Examples: 1 1 1 = =

y x ( y x) x y a b (b a) b a = = x y ( y x) y x

Simplifying Rational Expressions

As with the more familiar numerical fractions we can reduce rational expressions by canceling common factors in the numerator and the denominator.

pk =

p

qk q Examples;

Reduce x8

x12

12x 2 yReduce

9xy 2

4x 2 1Reduce

2x 2 + x

Warning: it is common to try factoring improperly. Remember, you can only factor out factors that in a product, not terms in and addition or subtraction

Proper reducing

Improper reducing

Addition and Subtraction of Rational Expressions

As in numerical fractions, we can only add and subtract rational expressions (algebraic fractions) if they have like (common) denominators.

a c a + c+ = b b b

a c a c = b b b

Examples: 2x 3x 2x + 3x 5x+ = = 7 7 7 7

If the two fractions do not have a common denominator, we must convert the individual fractions, separately, to fractions with the same denominator(LCD).

Example: 2x 4x

Add + , the denominators are 3 and 5, so the LCD is 15 3 5

2x 2x 5 10x 4x 4x 3 12x = = and = = therefore we get 3 3 5 15 5 5 3 15

10x 12x 22x+ = 15 15 15

Finding the Least Common Denominator We create a LCD in algebraic fractions just like we did in arithmetic fractions

1. Factor each denominator completely 2. List each prime factor the greatest number of times that it appears

in either of the factored forms of the denominator. 3. The product of these factors is the LCD.

Examples: 10x = 2*5* x7 8

+ 15x = 3*5* x 10x 15x LCD = 2*3*5* x = 30x

7 7 3 21 = = 10x 10x 3 30x

8 8 2 16 = = 15x 15x 2 30x

therefore 21 16 37+ =

30x 30x 30x

18x = 2 3 3 x5 7+ 12xy = 2 2 3 x y

18x 12xy LCD = 2 2 3 3 x y = 36xy

5 5 2y 10y= = 18x 18x 2y 36xy

7 7 3 21 = = 12xy 12xy 3 36xy

therefore 10y 21 10y + 21+ = 36xy 36xy 36xy

3 4 LCD = a 2 b3 c a 2 b b 3 c

3 3 b 2 c 3b 2 c = = 2 2 2 2 3a b a b b c a b c

do not simplify at this point

4 4 a 2 4a 2 = = 3 3 2 2 3b c b c a a b c

therefore 3b 2 c 4a 2

a 2 b3 c

1 4+ LCD = ( y + 3)(y + 4) y + 3 y + 4

1 1 y + 4 y + 4 = = y + 3 y + 3 y + 4 (y + 3)(y + 4)

4 4 y + 3 4(y + 3) 4y +12 = = = y + 4 y + 4 y + 3 (y + 3)(y + 4) ( y + 3)(y + 4)

therefore y + 4 4y +12 5y +16+ =

( y + 3)(y + 4) ( y + 3)( y + 4) (y + 3)(y + 4)

Multiplication and Division of Rational Expressions

Multiplication of Rational Expressions

Multiplication of Rational Expressions is primarily the same as multiplication with numerical fractions.

Examples:

Division of Rational Expressions

Division of Rational Expressions involves using the old adage flip the last guy and multiply Invert and Multiply

Examples: 4 3t = 4 7t = 28t = 28 25 7t 25 3t 75t 75

Note the inverting

3 3 2 3 2 216a 8a 16a 25b 400a b 10a = = = 3 2 3 35b 25b 5b 8a 40ab b

Complex Fractions

A complex fraction is one in which the numerator or denominator or both contain fractions.

Examples: 5 16

3 7 ,

11 x ,

9 x

are all complex fractions

x 3x

To simplify a complex fraction, we express it as a simple fraction of lowest terms. The best way to handle a complex fraction is to realize that fractions are really quotients. Then you can use the old adage Flip the last guy and multiply (Invert and multiply)

Examples: 3

3 3 3 5 15 54 = = = = 3 4 5 4 3 12 4 5

3 1+ 4 3 Hint: confront the top and bottom separately 1 1+ 3 6

3 1 9 4 13+ = + = 4 3 12 12 12

1 1 2 1 3 1+ = + = = 3 6 6 6 6 2

13 13 2 1312 = =

1 12 1 6 2

1 x

x remember to handle top and bottom separately 1

1+ x

1 x 1 x 2 1 x 2 1 x = = =

x 1 x x x x

1 1 1 x 1 x +11+ = + = + =

x 1 x x x x

x 2 1 x 2 1 x (x +1)(x 1)xx = = = x 1

x +1 x x +1 x(x +1) x

Equations involving Algebraic Fractions

We have to occasionally solve equations involving fractions. We have a procedure to follow to help us in this quest.

Solve equations involving fractions

1. Determine the LCD 2. Multiply both sides of the equation by the LCD (this step eliminates the

fractions) 3. Solve the resulting equation 4. Check you solution. Sometimes the solution to the resulting equation will not

be a solution to the original algebraic problem (it will cause the denominator of the fraction to be zero)

Examples:

t 2t 55+ = LCD = 12 4 3 12

t 2t 55

12( + = )

4 3 12 t 2t 55

12( ) +12( ) = 12( )4 3 12

3t + 8t = 55 11t = 55 t = 5

1 1 5 1+ = + , LCD = 6y y 2 6 y 3

1 1 5 1

6y( + = + )

y 2 6y 3

1 1 5 1

6y( ) + 6y( ) = 6y( ) + 6y( )

y 2 6y 3

6 + 3y = 5 + 2y 3y 2y = 5 6 y = 1 since y = -1 will not cause any of the denominators to become zero, it is the solution

w 2+ 2 = , LCD is w-2 w 2 w 2

w 2

(w 2)( + 2 = )

w 2 w 2 w 2

(w 2)( ) + (w 2)2 = (w 2)( ) w 2 w 2

w + 2w 4 = 2 3w = 6 w = 2 But w =2 causes the denominator of the fractions to become zero therefore it cannot be a solution. (There is no solution)

The Cartesian Coordinate System

The Cartesian Coordinate System (Rectangular Coordinate System) is the perpendicular crossing of two number lines (axis), one horizontal (x) and one vertical (y).

Using the Cartesian Coordinate System, we can assign names to points on a plane.

The point where the two axes cross is called the origin.

Origin

We name points on the Cartesian Plane by using the ordered pair (x,y), where x (abscissa) is the position along the horizontal (x) axis and y (ordinate) is the position along the vertical (y) axis.

(x,y) is called an ordered pair because, in this case, order matters.

The x- axis and the y-axis divide the Cartesian Plane into 4 sections called quadrants (QI, QII, QIII, QIV)

Points on either axis are not in any quadrant.

The graph of an equation is the set of points that are solutions to the equation 3x + 2y = 12 (4,0) is a solution (0,6) is a solution (1,1) is not a solution

Graphs of the Linear Equation

The graph of an equation of two variables is the set of all points that are solutions to the equation. (Obviously, it is impossible to plot all the points that are solutions to equation. We determine the type of graph that is associated with the specific type of equation and plot enough points to position the graph).

The Linear Equation

An equation of the form Ax + By + C = 0 is called a linear (first degree) equation.

The graph of a linear equation is a straight line.

To graph a linear equation 1. Find three points that solve the equation (pick 3 values for x and evaluate

the corresponding values for y). (Note: Technically you can draw a line using only two points, but we use three for accuracy in our rough sketches)

2. Plot the3 points on a Cartesian Plane. 3. Draw a straight line through the 3 points. (If you cannot draw a straight

line between the three points, you probably made a mistake in step 1)

Examples: Graph y = x 2

x y = x 2 Point -1 y = 1 2 = -3 (-1, -3) 0 y = 0 2 = -2 (0, -2) 2 y = 2 2 = 0 (2, 0)

Plot the points

Draw line through the points

Graph y = -2x + 4

x y = -2x + 4 Point 0 y = -2(0) + 4 = 4 (0,4) 1 y = -2(1) + 4 = 2 (1,2) 2 y =2(2) + 4 = 0 (2,0)

Special Lines

A linear equation with B = 0, Ax + C = 0 is represented by a vertical line through the point

Example: Graph 2x + 4 = 0

2x + 4 = 0 2x = -4 x = -2

A linear equation with A = 0, By + C = 0 represents a horizontal line through the point

Example: Graph 3y 12 = 0

3y = 12 y = 4

Intercept of a Line

All straight lines cross at least one of the coordinate axes. The points where the line crosses the axis is called the intercept.

The x-intercept is the point where line crosses the x-axis. To find the x-intercept of a line, set y = 0 and solve the resulting equation for x. (x,0)

The y-intercept is the point where the line crosses the y-axis. To find the y-intercept of a line, set x = 0 and solve the resulting equation. (0,y)

Example: Find the x-intercept and the y-intercept of 3x + 2y = 6

x-intercept y = 0 3x + 2(0) = 6

3x = 6 x = 2 (2,0)

y-intercept x = 0 3(0) + 2y = 6

2y = 6 y = 3 (0,3)

Special Cases

Vertical Lines (x = K) have only a x-intercept Example:

x = 2

Horizontal Lines (y = K) have only a y-intercept Example:

y = 3

Slope of a Line

We generally use the term slope to refer to the steepness of a line.

We also use this term when we discuss the steepness of such physical objects as the slope of a roof or a ski-slope.

In algebra, we define the slope of a line as the ratio of the change of Vertical distance (rise) to the change in Horizontal distance (run) between two points

rise y2 y1slope = m = = , where (x1,y1)and (x2,y2) run x2 x1

are two points on the line.

Examples: Find the slope of the line through the points (1,2) and (2,4)

4 2 2 m = = = 2

2 1 1

Find the slope of the line through (0,0) and (2,-4) 4 0 4

m = = = 2 2 0 2

Properties of Slope

Simultaneous Linear Equations

The solution to a linear equation in 2 variables (x,y) is a line, with and infinite number of solutions. If we have the need to solve two linear equations in 2 variables there are three possible scenarios.

First, the two lines could be parallel and have no points in common There are NO Solutions in this case

Second, the two lines meet in one point Then there is one solution to the system

Third, the two lines are actually the same line Then there is an infinite number of solutions

Solving Simultaneous Linear Equations (Substitution Method)

We can solve simultaneous linear equations (2 equations and 2 unknowns) by solving one of the equations for one of the variables and substituting into the other.

Example: 2x + 3y = 8 x + 2y = 5

We can solve the second equation (x + 2y = 5) for either x or y. It is simpler in this case to solve for x, so we get

x = 5- 2y

Now we can substitute this value for x into the first equation (2x + 3y = 8) and solve for y

2(5-2y) + 3y = 8 10 4y + 3y = 8

10 y = 8 10 y 10 = 8 10

-y = -2 y = 2

Now, we know y = 2, we can substitute this back into either equation to get x

2x + 3(2) = 8 2x + 6 = 8

2x + 6 6 = 8 6 2x = 2 x =1

Check: 2(1) + 3(2) = 2 + 6 = 8 and 1 +2(2) = 1 + 4 = 5

Therefore the point (1,2) is on both lines

Example: 5x + 2y = -7 3x + y = -5

Solve the bottom equation for y and get y = -5 3x Then substitute 5 3x in place of y in the top equation

5x + 2(-5 3x) = -7 5x 10 6x = -7

-10 x = -7 -x = 3 x = -3

Now we can solve for y, in the equation that says y = -5 3x and get y = -5 3(-3) = -5 + 9 = 4

So the point (-3,4) solve the system

Solving Simultaneous Linear Equations (Elimination Method)

A method for solving simultaneous linear equations is to eliminate one of the variables by adding or subtract the equations in the system.

There are a few rules that we need to keep track of We can multiply each of the equations by a constant (we do this to make the

coefficients of one the variables in both equations the same We can add the equations together to form a third equation that only has one

variable

Example: (no multiplication needed) x + y = 8 x y = 4

Notice that if we add the two equations together we eliminate the y.

x + y = 8 x y = 4

2x = 12 x = 6 now we put x = 6 into either equation to solve for y

6 + y = 8 y = 2

so the point (6,2) solves this system

Example: (multiplication in only one equation) 3x + 2y = 5 9x 4y = 5 if we multiply the top equation by 2 we get 6x + 4y = 10 and then addition will

eliminate the y 6x + 4y = 10 9x 4y = 5

15x = 15 x = 1

now, we can substitute x = 1 into either equation to get y

9(1) 4y = 5

9 4y = 5

-4y = -4

y = 1

therefore, (1,1) is the solution

Example: (multiplication in both equation) 4x 3y = 8 3x + 5y = -2

if we multiply the top by 5 and the bottom by 3 we can get the coefficients of y to be 15 and 15 which will add to 0 and eliminate y

20x 15y = 40 9x +15y = 6

29x = 34 34

x = 29

now we can substitute 34/29 into the first equation to solve for y

344( ) 3y = 8

29 136 3y = 8 29

136 3y = 8 29

232 136 3y = 29 29

96 3y = 29

32 y =

29 34 32

therefore the point , is the solution. 29 29

Dont let fractions frighten you, they appear occasionally and we have to deal with them

Introduction to Quadratic Equations

Many of the applications we encounter in the study of algebra involve solving quadratic equations of the form ax2 + bx + c = 0 .

Quadratic equations can have 0, 1 or 2 solutions in the real number system.

Test for the number of solutions of a quadratic equation

Define the discriminant (b2 4ac) 2If (b 4ac)> 0 , there are 2 solutions in the real number system

2If (b 4ac)= 0 , there is 1 solution in the real number system 2If (b 4ac)< 0 , there are no solutions in the real number system

Examples: 6x2 +5x 4 = 0

(b2 4ac) = 52 (4)(6)(-4) = 25-(-48) = 25 + 48 = 73 73 > 0 so there are 2 solutions

x2 2x + 1 = 0 (b2 4ac) = 22 (4)(1)(1) = 4 4 = 0 0 = 0 so there is 1 solution

x2 + 1 = 0 (b2 4ac) = 02 (4)(1)(1) = 0 4 = -4 -4 < 0 so there are no solutions

Solving Quadratic Equations by Factoring

If a quadratic expression can be factored, then it is quite simple to solve by using the factors and the fact that if a*b = 0 then either a = 0 or b = 0.

Examples: x2 5x + 6 = 0 (x 2)(x-3) = 0 x 2 = 0 or x 3 = 0 x = 2 or x = 3

two solutions

x2 2x + 1 = 0 (x 1)(x 1) = 0 x 1= 0 or x -1 = 0 x = 1 or x = 1

one solution

x2 = 36 x2 36 = 0 (x 6)(x + 6) = 0 x 6 = 0 or x + 6 = 0 x = 6 or x = -6

two solutions

6x2 + 5x = 4 6x2 + 5x 4 = 0 (2x 1)(3x + 4) = 0 2x 1 = 0 or 3x + 4 = 0 2x = 1 or 3x = -4 x = or x =-3/4

Solving Quadratic Equations by Extracting the Roots

Some quadratic equations can be written with a perfect square of a form of the variable on one side of the equal sign. We can use roots extraction to solve the equation (not as bad as it might sound, dentally)

Examples: x2 = 36 we can take the square root of both sides

x2 = 36 remember 36 = 6 x = 6 or x = -6

4u2 9 = 0 4u2 = 9

94 2 =u 2u = 3 or 2u = -3 u = 3/2 or u = -3/2

25x2 8 = 0

2 8 x = 25

8 8 x = or x =

25 25

2 2 2 2 x = or x =

5 5

2(4 y 5) 6 = 0 2(4 y 5) = 6

4y 5 = 6 or 4 y 5 = 6

4y = 6 + 5 or 4y = 6 + 5

6 + 5 6 + 5 y = or y =

4 4

Solving Quadratics by Completing the Square

If a quadratic equation is composed of a perfect square, then it is easy to solve by extracting the root. Using a technique called completing the square, we can transform any quadratic equation into one composed of a perfect square.

(Completing the square is not a popular method for solving quadratic equations, but the technique is useful in other applications, so discussing the method is important.)

Completing the square 2 2 2If we look at the pattern formed by the perfect square (a + b) = a + 2ab + b ,

we notice that the middle term is double the product of the square root of the end terms.

2 2 2In general, we will see the perfect square as (x + k) = x + 2xk + k , we can see 2 1 2the relationship between the last two terms. k = ( (2k) ) the last term is the square of

2 one-half the coefficient of x.

Using this relationship we can change any quadratic into a perfect square. Examples:

Change x2 + 10x into a perfect square x2+10x

x2 + 10x +52 52

(x2 + 10x + 25) - 25 (x + 5)2 25

x2 2x x2 2x + 12 12

(x2 2x + 1)2 1 (x 1)2 1

r2 + 7r

r2 +7r + 7 2 - 7 2 2 2

(r2 +7r + 494

) - 494

)2(r + 7 - 49 2 4

of 10 is 5 Add 52 to the problem, also subtract the same thing to maintain the integrity of the problem Factor the first part. Note: the + comes from the +10x and the 5 is the square root of the 25 in the parenthesis

of 2 is 1 Add 12 to the problem and also subtract 12

Factor the first part Note: the comes from the 2x and the 1 is the square root of the 1 in the parenthesis

of 7 is 7 , you could say 3.5 but that is 2

harder to work with Add 7 2 to the problem and also subtract it

2

off Factor the first part

Note: the + is from + 7 and the 7 is the 2 2

square root of 49 4

Solving Quadratic Equations using Completing the Square

1. Write the problem with variables on one side of the equal sign and constants on the other side

2. Divide both sides of the equation by the coefficient of the squared term (completing the square requires there to be a 1 in the first term)

3. Complete the square on the variable side 4. Add the new constant to both sides (eliminates it from the variable side) 5. Solve the equation by extracting the root

Examples: Solve x2 + 2x- 4 = 0

x2 + 2x- 4 = 0 Move 4 to other side x2 + 2x = 4 Complete the square

(x2 + 2x 1) - 1 = 4 Add 1 to both sides (x2 + 2x 1) = 5 Factor the variable side

(x + 1)2 = 5 Extract the root Solve equations (x+1) = 5 or (x + 1) = - 5

x = 5 - 1 or x = - 5 -1

Solve 2y2 12y 7 = 0 2y2 12y 7 = 0 Add seven to both sides

2y2 12y = 7 Divide both sides by 2 y2 6y = 7 Complete the square

2 7y2 6y + 32 32= 2

y2 6y + 9 9 = 7 Add 9 to both sides 2

y2 6y + 9 = 7 + 9 2

y2 - 6y + 9 = 25 Factor left side 2

(y 3)2 = 25 Extract the root 2

25 25 Add 3 to both sides y 3 = or y 3 = 2 2

Simplify y = 25 + 3 or y = - 25 +3 2 2

5 2 5 2 y = + 3 or y = - + 3

2 2

6 + 5 2 6 + 5 2 y = or y =

2 2

Solving Quadratic Equations using the Quadratic Formula

There are many techniques for solving quadratic equations. Completing the square will solve any quadratic equation, but it is not very efficient. The most efficient method for solving quadratic equations is to use the quadratic formula.

The Quadratic Formula

The standard quadratic equation has the form ax2 + bx + c = 0. The solution for this type of equation can be found by the following formula

2a

Examples: Solve 3x2 4x + 1 = 0

a = 3, b = -4, c = 1 therefore we get

(4) (4)2 4(3)(1) x =

2(3)

acbb x

42 =

4 16 12 x =

6

4 4 x =

6

4 2 x =

6

4 + 2 4 2 x = or x =

6 6

1 x = 1 or x =

3

Solve 2x2 + 3x 1 = 0 a = 2, b = 3, c = -1

3 32 4(2)(1) x =

2(2)

3 9 + 8 x =

4

3 17 x =

4

3 + 17 3 17 x = or x =

4 4

Complex Numbers

Most of the quadratic equations you will need to solve will have solutions in the real number system. But there are quadratic equations that do not have solutions in the real number system.

Example: x2 +1 = 0 x 2 = 1 x = 1

1 has no meaning in the real number system. To solve problems of this type, we introduce a new number system, the complex

number system.

Complex Number System Define 1 = i, so that we get x2 = -1 We then define a number a + bi where a, b are real numbers and i is defined as above.

We can now define the square root of negative numbers in the complex number system.

9 = 9 1 = 3i 121 = 121 1 = 11i

Examples of complex numbers

5 + 3i, 2 7i, 8, 4i are all complex numbers

Addition and Subtraction of Complex Numbers

The addition and subtraction of complex numbers follows the same rules as we use in algebra (combining like terms). Add and/or subtract the real parts and the complex parts separately.

Examples: (2 + 7i) + ( 3 4i) = (2 + 3) + (7 4)i = 5 + 3i

(6 + 5i) (3 7i) = (6 3) + (5 (-7)i = 3 + 12i

Multiplication of Complex Numbers

The multiplication of complex numbers follows the same rules of algebraic multiplication, using either the distributive property or FOIL.

Examples:

(2 + 3i)(1 + 7i) 2(1) + 2(7i) + 3i(1) + (3i)(7i) 2 + 14i + 3i + 21i2 note: i2 = -1 2 + 14i + 3i 21 -19 + 17i

(4 i)(3+ 2i) 4(3) + 4(2i) i(3) i(2i

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