# Chapter 3 Math

### Notes, Examples, and Practice

## Solving Equations Using Addition and Subtraction

**Example: x+3=10**

Move 3 to other side of equal sign to find x

Numbers switch signs ( - or + ) when they cross the equal sign

Subtract 3 from 10

**Example: x-5=4**

Follow rules of addition, but instead of subtracting, add to other side.

Add 5 to 4

**Word Problem:**

The temperature rose 15° to -7°F. What was the original temperature.

Write equation for problem ⇾ x+15=-7

Solve equation ⇾ x=-22°F

## Solving Equations Using Multiplication and Division

Multiply each side by the same nonzero number

X/2=3 X=6 (multiply by 2)

4X=12 X=3 (divide by 3)

You do the inverse operation in order to isolate X; If it is dividing, Multiply each side, and if it is multiplying, divide each side.

-4X=1 -write original equation

-4X/4= 1/-4 -divide each side by -4

X= -¼ -simplify

X/5= -30 - write original equation

5(X/5)= 5(-30) -multiply each side by 5

X= -150 -simplifyWhen it is a fraction- multiply by the reciprocal

10= -2/3X -write original equation

(-3/2) 10= (-3/2)(-2/3X) -multiply each side by -3/2

-15=m -simplify## Solving Multi-Step Equations

Simplify one or both sides of the equation (if needed)

- Use inverse operations to isolate variable
**Example 1 - Solving Linear Equations**Solve 1/3x+6=-8

1/3x+6=-8 Write original equation

1/3x+6=-8 Subtract 6 from each side

-6 -6

1/3x=-14 Simplify

3(1/3x)=3(-14) Multiply each side by 3

x=-42 Simplify**Example 2 - Combining Like terms**Solve 7x-3x-8=24

7x-3x-8=24 Write original equation

4x-8=24 Combine like terms

4x-8=24 Add 8 to each side

+8 +8

4x=32 Simplify

x=8

## Solving Equations with Variables on Both Sides

Doesn't matter which side variables are moved to, answer will be the same either way

**Example 1 -**

Solve 7x + 19 = -2x + 55

7x + 19 + 2x = -2x + 55 + 2x Add 2 to both sides

9x + 19 = 55 Simplify

9x + 19 - 19 = 55 - 19 Subtract 19 from both sides

9x = 36 Simplify

x=4**Word Problem:**

UPS charges $7 for the first pound, and $0.20 for each additional pound. FedEx charges $5 for the first pound and $0.30 for each additional pound. How many pounds, p, will it take for UPS and FedEx to cost the same?

7+0.2p = 5+0.3p

7+0.2p-0.2p = 5+0.3p-0.2p Subtract 0.2 from both sides

7 = 5+0.1p Simplify

7-5 = 5+0.1p-5 Subtract 5 from both sides

2 = 0.1p Simplify

20 = p## Linear Equations and Problem Solving

**Example - **

**# taking Spanish now + Rate of increase for Spanish x # of years = 3 ( # taking Spanish now - rate of decrease for French x # of years ) **

## Solving Decimal Equations

Three people want to share equally in the cost of a pizza. The pizza costs $12.89.

You can find what each person will have to pay by solving 3X= 12.89

**Example-**

3X= 12.89 -write original equation

X= 4.2966… -the exact answer is a repeating variable

X = 4.30 -round to the nearest cent.

## Formulas and Functions

**Formula-**an algebraic equation that relates two or more real-life quantities.

**Function- **a relationship or expression involving one or more variables.

**Example 1** **-**

A=lw Solve for length l.

A/w=l To get l by itself, divide each side by w

(Substitute given values into new formula)

35/7=5

**The length of rectangle is 5 feet.**

**Example 2 - **

C=5/9(F - 32) Solve for F

9/5 ⋅ C = 9/5 ⋅ 5/9(F - 32) Multiply each side by 9/5

9/5C = F - 32 Simplify

9/5C + 32 = F - 32 + 32 Add 32 to each side

**9/5C + 32 = F**

## Rates, Ratios, and Percents

**Unit Rate-**a rate per one given unit

**Example - **

60 miles / 3 hours Simplify

**20 miles / 1 hour**

**Ratio- **the quantitative relation between two amounts showing the number of times one value contains or is contained within the other

**Example - **

9:3 or 9 to 3 or 9/3 Simplify

**3:1 , 3 to 1, 3/1**

**Percent- **one part in every hundred

**Example - **

60% of 200 Simplify

**30% of 100**