Algebra 1: NAME: NOTES 1.1: Variables and Expressions 1/A1Ch1Notes.pdf · 1) Lou drives 65 mi/h. Write an expression for the number of miles that Lou drives in t hours 2) Miriam is

Algebra 1: NAME: __________________________

NOTES 1.1: Variables and Expressions

Definitions

Variable

Constant

Numerical Expression

Algebraic Expression

Mathematical Operations and Words

Example 1:

Give two ways to write each algebraic expression in words

x + 3 m – 7 2∙y k 5

You Try:

Give two ways to write each algebraic expression in words

4 – n 9 + q 3(h)

Example 2:

Translating from words to algebraic symbols

1) Eve reads 25 pages per hour. Write an expression for the number of pages she reads in h hours

2) Sam is 2 years younger than Sue, who is y years old. Write an expression for Sam’s age

3) William runs a mile in 12 minutes. Write an expression for the number of miles that William runs in m minutes.

You Try:

Translating from words to algebraic symbols

1) Lou drives 65 mi/h. Write an expression for the number of miles that Lou drives in t hours

2) Miriam is 5 cm taller than her sister, who is m cm tall. Write an expression for Miriam’s height in centimeters

3) Elaine earns $32 per day. Write an expression for the amount that she earns in d days

5

t

Definitions

Evaluating an Expression

Value of an Expression

Steps for Evaluating an Expression

STEP 1 STEP 2 STEP 3

Example 3:

Evaluate each expression for x = 8, y = 5, and z = 4

x + y

You Try:

Evaluate each expression for m = 3, n = 2, and p = 9

mn p – n p m

z

x

Example 4:

Approximately fourteen 20-ounce plastic drink bottles must be recycled to produce 1 square foot of carpet

◦ Write an expression for the number of bottles needed to make c square feet of carpet

◦ Find the number of bottles needed to make 40, 120, and 224 square feet of carpet

You Try:

To make one sweater, sixty-three 20-ounce plastic drink bottles must be recycled

◦ Write an expression for the number of bottles needed to make s sweaters

◦ Find the number of bottles needed to make 12, 25, and 50 sweaters

1.2 and 1.3 Note-Sheet: One – Step Equations

(Addition, Subtraction, Multiplication, and Division)

GOAL: to isolate the variable

What you do to one side of the

equal sign you must do to the

EXACT same thing to the

other side of the equal sign

Equations must balance

THINK OF OPPOSITES

Subtraction – Addition

Addition – Subtraction

Multiplication – Division

Division -- Multiplication

Inverse Operations

To solve one step equations, you need to

ask three questions about the equation:

• What is the variable?

• What operation is performed on

the variable?

• What is the inverse operation?

(The one that will undo what is

being done to the variable)

Solve using the inverse operation

Helpful Hints

* Work on the side with the variable

* Get rid of ALL fractions first by

multiplying EVERY term by the

denominator

* You can check your solution by

substituting your answer back into the

original problem and evaluating (both

sides of the equal sign should be the

same if your solution is correct)

* Other (Do you have any hints)?

Example 1:

x – 10 = 4

1) What is the variable? ___________

2) What operation is being performed on the variable? ____________________

3) What is the inverse operation? ______________________________

Solve showing ALL work!!!

x – 10 = 4

You Try:

-5 = k + 5





-5 = k + 5

Example 2:

7x = 56





7x = 56

You Try:

54

k





54

k

Example 3:

359

5v


359

5v

You Try:

r8

39


r8

39

Example 4:

A person’s maximum heart rate is the highest rate, in beats per minute, that the person’s heart should reach. One method to estimate

maximum heart rate states that your age added to your maximum heart rate is 220. Using this method, write and solve an equation to find the

maximum heart rate of a 15-year-old

Define the variable

r = maximum heart rate

Key Phrases in the problem and what they mean.

1) age added to maximum heart rate:

2) is 220:

3) age:

Equation:

Solution:

You Try:

Over 20 years, the population of a town decreased by 275 people to a population of 850. Write and solve and equation to find the original

population.

Define the variable = _______________________________________

Key Phrases in the problem and what they mean.

Equation:

Solution:

What??? I just learned 1-step!

Relax. You’ll use what you already know

to solve 2-step equations.

1.4 Note-Sheet: SOLVING EQUATIONS

(Notes: 3 parts: two-step equations, fractions, distribute and combine like terms)

Two – Step Equations

AND

Equations With Fractions

Two – Step Equations

STEPS: Follow all steps below to solve two-step equations (GOAL: get the variable by itself)

1st = Work on the side with the variable. Add or subtract (this is the number without the variable)

2nd = Divide last (this is the number with the variable)

Example 1 Steps Taken

5q – 7 = 13

Example 2 Steps Taken You Try Steps Taken

8x + 5 = 61 3a – 8 = 4

Example 3 Steps Taken You Try Steps Taken

-47 = 3x – 50 -35 = 2p + 10

Equations with FRACTIONS!!!

STEPS: Follow all steps below to solve equations with fractions (GOAL: get the variable by itself)

1st = If you see fractions; get RID of them FIRST by multiplying EVERY term by the denominator

2nd = Work on the side with the variable. Add or subtract (this is the number without the variable)

3rd = Divide last (this is the number with the variable)

Example 1 Steps Taken Example 2 Steps Taken

208

5y 157

3

2

x

You Try Steps Taken

145

10 m

EQUATIONS:

Distributive Property and Combining Like Terms

STEPS: When ALL variables are on the same-side of the equal sign.

1st = If you see parenthesis ( ) then distribute first

2nd = Combine any like terms on the same-side of the equal sign

3rd = Work on the side with the variable. Add or Subtract (this is the number without the

variable)

4th = Divide last (this is the number with the variable)

EXAMPLE #1 12 – x = -5

GOAL: Get the "x" by itself

You TRY. . .

12 – x = -5 Steps Taken 21 = -x – 3 Steps Taken

EXAMPLE #2 4x + 2 + 3x = 58

GOAL: Get the "x" by itself

You TRY. . .

4x + 2 + 3x = 58 Steps Taken 10x + 2 – 5x = 52 Steps Taken

EXAMPLE #3 4(x + 12) + 7x = 26

You TRY. . .

4(x + 12) + 7x = 26 Steps Taken -2(x – 3) = -12 Steps Taken

EXAMPLE #4

The price of an automobile tire has been reduced by $15. The cost of a set of 4 tires at the reduced price is

$600. Write and solve an equation to find the original cost of a tire.

Define the variable.

X = cost of tire

Key phrases in the problem and what they mean.

1) “reduced” by $15 =

2) “set of 4 tires” =

3) “the reduced price is $600” =

4) Equation: _______________________________________

5) Solution = ________________________________________

You Try

The height of an ostrich is 20 inches more than 4 times the height of a kiwi. Write and solve an equation to find the

height of a kiwi.

Define the variable

Identify key phrases in the problem and what they mean

Equation

Solution

1.5 Note-Sheet: Variables on Both Sides:

STEPS: Follow all steps below to solve equations

1st = If you see parenthesis ( ) then distribute first

2nd = Combine any like terms on the same-side of the equal sign

3rd = Add or subtract to move variable terms to one side of the equal sign (Variables on

one side, numbers on the other side)

4th = Work on the side with the variable. Add or Subtract (this is the number without

the variable)

5th = Divide last (this is the number with the variable)

EXAMPLE #1 You Try #1

4x = 2x + 6 Steps Taken -3x + 15 = 2x Steps Taken


-4a + 6 = 2a – 36 Steps Taken 10w + 300 = 20w – 50 Steps Taken


-4(x – 4) = 6x + 36 Steps Taken 10x – 2 = 2(x + 7) Steps Taken

1.7 Note-Sheet: Solving Absolute-Value Equations

Absolute Value (of x)

Symbol |x|

The distance x is from 0 on a number line

Always positive

STEPS for Solving

1st = Get the | | symbol by itself (isolate it)

2nd = Set up two cases (one positive and one negative) NOTE: get rid of the | |

|x| = a

Case 1: x = a Case 2: x = -a

3rd = Solve each case

***HINTS:

1) 0 does not have and opposite (|0| = 0)

2) | | ≠ negative number (|x| ≠ -2)

Example: |x| = 5

What are the possible values of x?

Set up 2 cases and solve

Case 1 Case 2

You Try: |x| = 4

What are the possible values of x?

Set up 2 cases and solve

Case 1 Case 2

EXAMPLE: 4|x + 2| = 24 You Try: 2|x – 1| = 4

Case 1 Case 2 Case 1 Case 2

EXAMPLE: |x + 3| + 4 = 4 You Try: 5 = |x + 2| + 8

Case 1 Case 2 Case 1 Case 2

Documents

Algebra 1: NAME: NOTES 1.1: Variables and Expressions 1/A1Ch1Notes.pdf · 1) Lou drives 65 mi/h. Write an expression for the number of miles that Lou drives in t hours 2) Miriam is