6.2 Solving Linear Equations

Objective: To solve linear equations

Linear equations• An equation is made up of two algebraic

expressions put together with an = sign.• To solve an equation means to find the

value of the variable that makes the statement true.

• For example, x = 2 is a solution to x + 3 = 5 because 2 + 3 = 5 is a true statement.

• Linear equations can be written in the form ax + b = c.

Solving equations by addition and subtraction

• If a number is being added to the variable subtract it from both sides

• If a number is being subtracted from the variable add it to both sides

Examples

• x + 10 = 23

• x - 5 = -18

• 1 + x = 2

Solving equations using multiplication and division

• If the number is being multiplied times the variable then divide both sides by the number

• If the number is dividing the variable then multiply both sides by the number.

• If the variable is being multiplied times a fraction then multiply both sides by the reciprocal of the fraction.

Examples

• 8k = 36

• -7t = 49

• c/4 = 16

• 3/5 x = 15

Solving multi-step equations

• When solving multi-step equations, reverse addition and subtraction before multiplication and division.

Examples: Solve

1. 4a – 5 = 15

2. 9 65

y

23. 6 14

9v

24. 7

3

d

3b+15. 25

2

Solving equations with variables on both sides

1. 6k-3=2k+13

2. 9t+7=3t-5

3. 3n+1=7n-5

More examples: first use the distributive property

1. 2(x – 4) + 5x = -22

2. 8 – 2(t + 1) = -3t + 1

3. 5 + 2(k+4)=5(k – 3) + 10

4. 8x – 5(2 + x) = 2(x + 1)

Equations with fractions

• To solve equations with fractions first multiply both sides by the least common denominator of all denominators on both sides.

Examples: Equations with fractions

2 33. 9 7

5 5n n

1 15. 8 - 7

2 4p p

3 14. 4 6

4 7t t

3 391.

2 5 5

x x

2 51.

4 3 6

x x

Solving for a specific variable

• Solve T = D + pm for m

• Solve I = Prt for P

• Solve 3x + 6y = 12 for y

HW: p.289/1-73