Chapter 2

Solving Linear Equations

2.1 Writing Equations

4 Steps to Problem Solving– Explore the problem (read the whole thing)– Plan the solution (write the equation)– Solve the problem– Check the solution (does it make sense?)

How to use a Formula– Write the formula– Substitute for any known

variables– Solve the equation– Check the answer (does it

make sense?)

Perimeter=2l+2w Area of a Square= lw Area of a Triangle= ½ bh Area of a Circle= r2

2.2 Solving Equations by Using Addition and Subtraction

To solve an equation means to find all values of the variable that make it true

Addition Property of Equality: – If you add the same amount to each side of the

equal sign, the equation is true

Ex: m – 48 = 29

Subtraction Property of Equality:– If you subtract the same amount from each side,

the equation is true

42 + d = 27 n + 5 = 40

2.3 Solving Equations by Using Multiplication and Division

Multiplication Property of Equality:– If you multiply each side of the equal sign by the

same number, the equation is true

t/3 = 7 9/4g = 1/2

Division Property of Equality:– If you divide each side of the equal sign by the

same number, the equation is true

13s = 195 -3x = 36

2.4 Solving Multi-Step Equations

Add or Subtract the number farthest from the variable

Multiply or divide the number next to the variable

Simplify Check your answer

Examples:

(p – 15)/9 = -6 2/3y – 25 = 115

Even consecutive and odd consecutive numbers = x, x+2, x+4, x+6….

Consecutive numbers = x, x+1, x+2, x+3…

– Find three consecutive integers whose sum is 21– Find three consecutive even integers whose sum

is -42

2.5 Solving Equations with Variables on Each Side

Distribute and/or combine like terms Add or subtract the variables to one side

(move the smaller one) Add or subtract the numbers to the other side Solve as normal

Examples:

-2 + 10k = 8k -1 2m = 5 = 5(m – 7) -3m

2.6 Ratios and Proportions

Ratio: a comparison of two numbers by division– x to y x : y

ex: Your class has 21 students, 9 are boys and 12 are girls

a. ratio of boys to girls ________________

b. ratio of students to boys _____________

Scale: a ratio that shows that a model is proportional to an actual object

y

x

Proportion: shows that two ratios are equal– a and d are the extremes– b and c are the means

Solve a proportion by cross multiplying– .– ad = bc

d

c

b

a

d

c

b

a

Ex: Determine if it is a proportion.

Ex: Solve the proportion.– a.

– b.

42

35

36

15 16

24

15

n

9

12

6

2

w

2.7 Percent of Change

New # is greater than original # = % of increase New # is less than original # = % of decrease

To solve: %100#

original

originalnew

Original: $25

New: $28

Original: 16

New: 3

Concert tickets cost $45 each. The tax is 6.25%. What is the total cost for one ticket?

A sweater is on sale for 35% off. The original price is $38. What is the sale price?

2.8 Solving for a Specific Variable

Use the normal order of operations and problem solving steps to get the specific variable on one side of the equal sign and everything else on the other side of the equal sign

Solve 3x – 4y = 7 for y. Solve 2m – t = sm + 5 for m.

Solve C=2 r for r Solve for a. 2

2

1ats

2.9 Weighted Averages

Weighted average: the sum of a product of units and value per unit, divided by the sum of the # of units

How many pounds of mixed nuts selling for $4.75 per pound should be mixed with 10 pounds of dried fruit selling for $5.50 per pound, to obtain trail mix that sells for $4.95 per pound?

# Units Price per Unit

Product

Dried Fruit 10 5.50 550

Nuts 4.75

Trail Mix 4.95

55.0 + 4.75x = 4.95(10 + x)SOLVE FOR X

x

10 + x

4.75x

4.95(10 + x)

An experiment calls for 30% solution of copper sulfate. Kendra has 40ml of 25% solution. How many ml of 60% solution should be added?

Amount/Units Product

25% Solution

60% Solution

30% Solution

40

x

40 + x

40 x .25= 10

.6x

.3(40 + x)

10 + .6x = .3(40 + x)

SOLVE FOR X