Algebraic Expressions 2x + 3y - 7

Algebraic Expressions

2x + 3y - 7What are the Terms?

2x + 3y - 7

2x + 3y - 7What are the variables?

2x + 3y - 7

Variables

2x + 3y - 7What are the coefficients?

2x + 3y - 7

Coefficients

2x + 3y - 7What is the constant?

2x + 3y - 7Constant

Algebraic ExpressionsPolynomial:monomial → x, 2xy, 4, 3x²y, … single termbinomial → x+1, 2xy+x, 3x²y+4, …two termstrinomial → 2x+3y+7, 3x²y+xy+4x, …three termspolynomial → …four or more terms

What is the area of a rectangle?

Length times WidthIf the length is 3 meters and the width is 2

meters, what is the area?A = L x W

A = 3 x 2 = 6 meters2

A, L and W are the variables. It is any letter that represents an unknown number.

An algebraic expression contains:

1) one or more numbers or variables, and

2) one or more arithmetic operations. Examples:

x - 33 • 2n4 1m

In expressions, there are many different ways to write multiplication.

1) ab 2) a • b 3) a(b) or (a)b 4) (a)(b) 5) a x b

We are not going to use the multiplication symbol any more. Why?

Division, on the other hand, is written as:

2) x ÷ 3

Here are some phrases you may have see throughout the year. The terms with * are

ones that are often used.Addition Subtraction Multiplication Divisionsum* difference* product* quotient*increase decrease times dividedplus minus multiplied ratioadd subtractmore than less thantotal

Write an algebraic expression for

1) m increased by 5.m + 5

2) 7 times the product of x and t.7xt or 7(x)(t) or 7 • x • t

3) 11 less than 4 times a number.

4n - 11

4) two more than 6 times a number.

6n + 2

5) the quotient of a number and 12.

Which of the following expressions represents 7 times a number decreased by 13?

1. 7x + 132. 7x - 133. 13 - 7x4. 13 + 7x

Which one of the following expressions represents 28 less than three times a number?

1. 28 - 3x2. 3x - 283. 28 + 3x4. 3x + 28

Write a verbal expression for:

1) 8 + a.

The ratio of m to rDo you have a different way of writing these?

The sum of 8 and a2)

Which of the following verbal expressions represents 2x + 9?

1. 9 increased by twice a number2. a number increased by nine3. twice a number decreased by 94. 9 less than twice a number

Which of the following expressions represents the sum of 16 and five times a number?

1. 5x - 162. 16x + 53. 16 + 5x4. 16 - 5x

Which of the following verbal expressions represents x2 + 2x?

1. the sum of a number squared and twice the number

2. the sum of a number and twice the number

3. twice a number less than the number squared

4. the sum of a number and twice the number squared

Which of the following expressions represents four less than the cube of a number?

1. 4 – x3

2. 4 – 3x3. 3x – 44. x3 – 4

Evaluate.21 222 2 • 2 = 423 2 • 2 • 2 = 82n7 We can’t evaluate because

we don’t know what n equals to!!

Competition Problems

Evaluating Algebraic Expressions

Evaluate the following algebraic expression using

m=7, n=8

n² - m

Answer:

x=5, y=2

8(x-y)

Answer:

x=7, y=2

yx ÷ 2

Answer:

x=1, z=19

z + x³

Answer:

m=3, p=10

15-(m+p)

Answer:

a=9, b=4

b(a+b) + a

Answer:

m=3, p=4

p²÷4-m

Answer:

x=4, y=2

y(x-(9-4y))

Answer:

x=9, y=1

x-(x-(x-y³))

Answer:

h=9, j=8

j(h-9)³ +2

Answer:

Simplifying Algebraic Expressions

REVIEW

Vocabularytermcoefficientlike terms

Insert Lesson Title Here

The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.

4x – 3x + 2

Like terms Constant

A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.

1x2 + 3x

Coefficients

In the expression 7x + 5, 7x and 5 are called terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by + and –.

7x + 5 – 3y2 + y + x3

term term term term

In the term 7x, 7 is called the coefficient. A coefficient is a number that is multiplied by a variable in an algebraic expression. A variable by itself, like y, has a coefficient of 1. So y = 1y.

Coefficient Variable

Coefficient

4a 23 3k2 x2 x

9 4.7t

4 23 3 1 1

Like terms are terms with the same variable raised to the same power. The coefficients do not have to be the same. Constants, like 5, , and 3.2, are also like terms.1

Like Terms

Unlike Terms

3x and 2x

5x2 and 2xThe exponentsare different.

3.2 and nOnly one term

contains avariable

6a and 6bThe variablesare different

w and w7 5 and 1.8

Identify like terms in the list.

Additional Example 1: Identifying Like Terms

3t 5w2 7t 9v 4w2 8v

Look for like variables with like powers.

3t 5w2 7t 9v 4w2 8v

Like terms: 3t and 7t, 5w2 and 4w2, 9v and 8v

Use different shapes or colors to indicate sets of like terms.Helpful Hint

Insert Lesson Title HereIdentify like terms in the list.

2x 4y3 8x 5z 5y3 8z

Look for like variables with like powers.

Like terms: 2x and 8x, 4y3 and 5y3 , 5z and 8z

2x 4y3 8x 5z 5y3 8z

Insert Lesson Title Here

Combining like terms is like grouping similar objects.

x x x x

x x x x x

4x + 5x = 9x

To combine like terms that have variables, add orsubtract the coefficients.

Using the Distributive Property can help you combine like terms. You can factor out the common factors to

simplify the expression.

7x2 – 4x2 = (7 – 4)x2

= (3)x2

Factor out x2 from both terms.

Perform operations in parenthesis.

Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Simplify the expression by combining like terms.

72p – 25p

72p and 25p are like terms.

Subtract the coefficients.

A variable without a coefficient has a coefficient of 1.

Write 1 as .

Add the coefficients.

and are like terms.

0.5m + 2.5n

0.5m and 2.5n are not like terms.

Do not combine the terms.

Simplify by combining like terms.

16p + 84p16p + 84p

16p + 84p are like terms.

Add the coefficients.

–20t – 8.5t2

–20t – 8.5t2 20t and 8.5t2 are not like terms.

–20t – 8.5t2 Do not combine the terms.

3m2 + m3 3m2 and m3 are not like terms.3m2 + m3

Do not combine the terms.3m2 + m3

Simplify 14x + 4(2 + x)

14x + 4(2) + 4(x) Distributive Property

Multiply.

Commutative Property

Associative Property

Combine like terms.

14x + 8 + 4x

(14x + 4x) + 8

14x + 4x + 8

18x + 8

14x + 4(2 + x)1.

Procedure Justification

6(x) – 6(4) + 9 Distributive Property

Multiply.

Combine like terms.

6x – 24 + 9

6x – 15

6(x – 4) + 91.

Simplify 6(x – 4) + 9. Justify each step.

–12x – 5x + x + 3a Commutative Property

Combine like terms.–16x + 3a

–12x – 5x + 3a + x1.

Simplify −12x – 5x + 3a + x. Justify each step.

Simplify each expression.

165 +27 + 3 + 5

Write each product using the Distributive Property. Then simplify.

5($1.99)

5($2) – 5($0.01) = $9.95

6(10) + 6(3) = 78

Simplify each expression by combining like terms. Justify each step with an operation or property.

301x – x

24a + b2 + 3a + 2b2

27a + 3b2

14c2 – 9c 14c2 – 9c

Let’s work more problems…

Simplify the following algebraic expression:

-3p + 6p

Answer:

7x - x

Answer:

-10v + 6v

Answer:

5n + 9n

Answer:

b - 3 + 6 - 2b

Answer:

-b + 3

10x + 36 - 38x - 47

Answer:

-28x - 11

10x-w+4y-3x+36-38x-47+32x+2w-3y

Answer:

w+x+y-11

Simplify the following algebraic expression using the distributive property:

6(1 – 5m)

Answer:

6 – 30m

-2(1 – 5v)

Answer:

-2 + 10v

-3(7n + 1)

Answer:

-21n - 3

(x + 1) 14∙

Answer:

14x + 14

(3 - 7k) (-2)∙

Answer:

-6 + 14k

-20(8x + 20)

Answer:

-160x - 400

(7 + 19b) -15∙

Answer:

-105 – 285b

Variable Expressions

))()((

))()()()(()(

yyymeansy

xxxxxtionmultiplicaforsparentheseusemeansx

Simplify:

(-a)²

Answer:

Substitution and EvaluatingSTEPS

1. Write out the original problem.2. Show the substitution with parentheses.3. Work out the problem.

3;4: xxifSolveExample 3;4 xxifSolve

3)4( = 64

Evaluate the variable expression when x = 1, y = 2, and w = -3

22 )()( yx

22 )2()1(

Step 1

Step 2

Step 3

2)( yx

2)2()1(

9)3( 2

Step 1

Step 2

Step 3

2)1)(3(

Step 1

Step 2

Step 3

3)1)(3(

Contest Problem

Are you ready?3, 2, 1…lets go!

Evaluate the expression when a= -2

a² + 2a - 6

Answer:

Evaluate the expression when x= -4 and t=2

x²(x-t)

Answer:

Evaluate the expression when y= -3

(2y + 5)²

Answer:

MULTIPLICATION PROPERTIESPRODUCT OF POWERS

This property is used to combine 2 or more exponential expressions with the SAME base.

53 22 )222( )22222( 82 256

))(( 43 xx ))()(( xxx ))()()(( xxxx 7x

MULTIPLICATION PROPERTIESPOWER OF PRODUCT

This property combines the first 2 multiplication properties to simplify exponential expressions.

2)56( )5()6( 22 9002536

3)5( xy ))()(5( 333 yx33125 yx

532 )4( xx 5323 ))(4( xx 5222 ))()(()64( xxxx

56 ))(64( xx 1164x

Problems

Simplify. Your answer should contain only positive exponents.

2n⁴ · 5n ⁴

Answer:

10n⁸

6r · 5r²

Answer:

6x · 2x²

Answer:

6x² · 6x³y⁴

Answer:

36x⁵y⁴

10xy³ · 8x⁵y³

Answer:

80x⁶y⁶

MULTIPLICATION PROPERTIESPOWER TO A POWER

This property is used to write and exponential expression as a single power of the base.

32 )5( )5)(5)(5( 222 65

42 )(x ))()()(( 2222 xxxx 8x

MULTIPLICATION PROPERTIESSUMMARY

PRODUCT OF POWERSbaba xxx

POWER TO A POWER baba xx

POWER OF PRODUCTaaa yxxy )(

ADD THE EXPONENTS

MULTIPLY THE EXPONENTS

Problems

(a²)³

Answer:

(3a²)³

Answer:

27a⁶

(x⁴y⁴)³

Answer:

x¹²y¹²

(2x⁴y⁴)³

Answer:

8x¹²y¹²

(4x⁴ x⁴)³∙

Answer:

64x²⁴

(4n⁴ n)²∙

Answer:

16n¹⁰

ZERO AND NEGATIVE EXPONENTSANYTHING TO THE ZERO POWER IS 1.

)2(1)2(

813131

DIVISION PROPERTIES QUOTIENT OF POWERS

This property is used when dividing two or more exponential expressions with the same base.

))()(())()()()((

xxxxxxxx

1))(( xxx

3 11111

DIVISION PROPERTIESPOWER OF A QUOTIENT

Hard Example

)3()2(

ZERO, NEGATIVE, AND DIVISION PROPERTIES

Zero power 1)( 0 x

Negative Exponents

Quotient of powers

Power of a quotient

Problems

3r³2r

Answer:

3xy 5x²( )

Answer:

9y²25x²

18x⁸y⁸ 10x³

Answer:

9x⁵y⁸5

Simplify:

(x⁴y¯²)(x¯¹y⁵)

Answer:

x³y³

8x (6x + 6)∙

Answer:

48x² + 48x

7n(6n + 3)

Answer:

42n² + 21n

2(9x – 2y)

Answer:

18x – 4y

1 + 7(1 – 3b)

Answer:

8 - 21b

-2 - 7(-1 – 3b)

Answer:

5 + 21b

3n(n² - 6n + 5)

Answer:

3n³ - 18n² + 15n

2k³(2k² + 5k - 4)

Answer:

4k⁵ +10k⁴ - 8k³

9(x² + xy – 8y²)

Answer:

9x² + 9xy – 72y²

9v²(u² + uv - 5v²)

Answer:

9v²u² +9v³u – 45v⁴

3x(5x+2) - 14(2x²-x+1)

Answer:

-13x² + 20x - 14

Simplify completely:

4x²y2x

Answer:

y¯¹ y¯²

Answer:

16x⁴y¯¹4x²y¯²

Answer:

36x³y⁶z¹²4x¯¹y³z¹⁰

Answer:

9x⁴y³z²

21x³y⁷z¹⁴ 30x³z¯⁵18x⁴y⁶ y¹²z¯⁶·

Answer:

35x²z¹⁵y¹¹

Algebraic Expressions 2x + 3y - 7

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