2.4 Solving Multi-Step

Equations

2.5 Solving Equations

with variables on both

sides

Remember solving an

equation is a balancing act.

What you do

to one side

you have to

do to the

other!!

I. Multi-Step Equations

A. Steps

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

2. Use inverse operations to isolate the variable. (DO THE OPPOSITE OF ORDER OF OPERATIONS)

To simplify you use:

To solve you do the opposite:

G E M/D A/S

G E A/S M/D

B. Solving a Linear Equation

863

1x Write the original equation.

66 Subtract 6 from each side.

143

1x

Simplify.

143

1

x

Multiply each side by 3.

42x Simplify. CHECK

3 x x 3

C. Combining Like Terms First…

24837 xx Write the original equation.

2484 x Combine like terms.

88 Add 8 to each side.

324 x Simplify.

8x Simplify.

CHECK

324 x

Divide each side by 4. 4 4

D. Using the Distributive Property…

28)4(35 xx Write the original equation.

281235 xx Distribute the 3.

28128 x Combine like terms.

Subtract from both sides.

2x

Divide both sides.

CHECK

1212

Simplify 168 x

8 88

Simplify.

E. Distributing a Negative…

21)2(34 xx Write the original equation.

21634 xxDistribute the 3 and the negative.

216 x Combine like terms.

Subtract from both sides.

5x

CHECK

66

Simplify

F. Multiplying by a Reciprocal First…

Write the original equation.

Multiply by the reciprocal.

Subtract 3.

Practice…

573

13

27

20132

1572

xx

x

x

x

6)2(12

18)2(3

947

x

x

xx

II. Solving Equations with

Variables on Both Sides

Solve 4x + 6 = x

Get all variables on one side.

Try to keep the variable positive.

Ex. 1: Solve 4x + 6 = x

4x + 6 = x

– 4x – 4x

6 = –3x

To collect the variable terms on one side,

subtract 4x from both sides.

Since x is multiplied by -3, divide both

sides by –3.

–2 = x

6 –3

–3x –3

=

Ex 2: Solve 9b – 6 = 5b + 18

9b – 6 = 5b + 18

– 5b – 5b

4b – 6 = 18

4b 4

24 4

=

To collect the variable terms on one side,

subtract 5b from both sides.

Since b is multiplied by 4, divide both

sides by 4.

b = 6

+ 6 + 6

4b = 24

Since 6 is subtracted from 4b, add 6

to both sides.

Ex 3: Solve 9w + 3 = 9w + 7

3 ≠ 7

9w + 3 = 9w + 7

– 9w – 9w To collect the variable terms on

one side, subtract 9w from both

sides.

There is no solution. There is no number that can be

substituted for the variable w to make the equation

true.

if the variables in an equation are eliminated and the

resulting statement is false, the equation has no solution.

If the resulting statement is TRUE, then the solution is “all

real numbers”.

Helpful Hint

To solve more complicated equations, you may need to

first simplify by combining like terms or clearing

fractions. Then add or subtract to collect variable terms

on one side of the equation. Finally, use properties of

equality to isolate the variable.

Ex 4: Solve 10z – 15 – 4z = 8 – 2z – 15

10z – 15 – 4z = 8 – 2z – 15

+ 15 +15

6z – 15 = –2z – 7 Combine like terms.

+ 2z + 2z Add 2z to both sides.

8z – 15 = – 7

8z = 8

z = 1

Add 15 to both sides.

Divide both sides by 8. 8z 8 8 8

=

Multiply by the LCD, 20.

4y + 12y – 15 = 20y – 14

16y – 15 = 20y – 14

Combine like terms.

y

5 3

4

3y

5

7

10 + – = y –

y

5 3

4

3y

5

7

10 + – = y –

Ex 5 (Fraction Busting)

Add 14 to both sides.

–15 = 4y – 14

–1 = 4y

+ 14 + 14

–1 4

4y 4

= Divide both sides by 4.

–1 4

= y

16y – 15 = 20y – 14

– 16y – 16y Subtract 16y from both sides.

Lesson Quiz

Solve.

1. 4x + 16 = 2x

2. 8x – 3 = 15 + 5x

3. 2(3x + 11) = 6x + 4

4. x = x – 9

5. An apple has about 30 calories more than an orange. Five oranges have about as many calories as 3 apples. How many calories are in each?

x = 6

x = –8

no solution

x = 36 1 4

1 2

An orange has 45 calories. An apple has 75

calories.