Multi-Step Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Multi-Step Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Solving equations that require more than one inverse operationβ€”typically involving distributing, combining like terms, and moving variables to one side before isolating the variable.

A one-step equation is like unwrapping one layer of packaging. A multi-step equation has several layers: first simplify each side (distribute, combine like terms), then peel off operations one at a time until xx stands alone. Think of it as cleaning up a messy room before finding what you're looking for.

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Distribute and combine like terms first, then undo operations one at a time until the variable is alone.

Common stuck point: The procedure for multi-step equations is the easy part; the trap is distributing only to the first term. Asking "Does isolating xx take more than one step β€” distributing, combining, or moving variables first?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does isolating xx take more than one step β€” distributing, combining, or moving variables first?

Worked Examples

Example 1

easy
Solve 3(x+2)βˆ’4=143(x + 2) - 4 = 14.

Answer

x=4x = 4

First step

1
Distribute: 3x+6βˆ’4=143x + 6 - 4 = 14.

Full solution

  1. 2
    Simplify: 3x+2=143x + 2 = 14.
  2. 3
    Subtract 2: 3x=123x = 12.
  3. 4
    Divide by 3: x=4x = 4.
Multi-step equations require multiple inverse operations. Distribute first, then combine like terms, then isolate the variable.

Example 2

medium
Solve x+12βˆ’xβˆ’34=3\frac{x+1}{2} - \frac{x-3}{4} = 3.

Example 3

medium
Solve 4(xβˆ’3)+2x=184(x - 3) + 2x = 18.

Example 4

medium
Solve 2x+13=xβˆ’22\frac{2x + 1}{3} = \frac{x - 2}{2}.

Example 5

medium
Identify whether 3(x+2)=3x+63(x + 2) = 3x + 6 has one, no, or infinitely many solutions.

Example 6

hard
Solve 3(2xβˆ’1)βˆ’4(x+2)=5(xβˆ’3)+63(2x - 1) - 4(x + 2) = 5(x - 3) + 6.

Example 7

hard
A rectangle has perimeter 4444 cm. Its length is 44 more than twice its width. Find the dimensions.

Example 8

challenge
A father is three times his son's age. In 1010 years, the father will be twice the son's age. Find their current ages.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve 2(xβˆ’5)=82(x - 5) = 8.

Example 2

hard
Solve 5βˆ’2(3x+1)=4xβˆ’75 - 2(3x + 1) = 4x - 7.

Example 3

easy
Solve 2x+3=112x + 3 = 11.

Example 4

easy
Solve 5xβˆ’7=185x - 7 = 18.

Example 5

easy
Solve 3(x+2)=153(x + 2) = 15.

Example 6

easy
Solve x4+1=5\frac{x}{4} + 1 = 5.

Example 7

easy
Solve 2x+5=x+92x + 5 = x + 9.

Example 8

easy
Solve 4xβˆ’2=2x+64x - 2 = 2x + 6.

Example 9

easy
Solve 7=3xβˆ’57 = 3x - 5.

Example 10

easy
Solve 2x3=8\frac{2x}{3} = 8.

Example 11

medium
Solve 3(2xβˆ’1)=4x+73(2x - 1) = 4x + 7.

Example 12

medium
Solve 5xβˆ’2(xβˆ’3)=185x - 2(x - 3) = 18.

Example 13

medium
Solve x+12+x3=4\frac{x+1}{2} + \frac{x}{3} = 4.

Example 14

medium
Solve 2(x+4)=3(xβˆ’1)2(x + 4) = 3(x - 1).

Example 15

medium
Solve 6x+2=2(3x+1)6x + 2 = 2(3x + 1).

Example 16

medium
Solve 4(xβˆ’2)=4x+14(x - 2) = 4x + 1.

Example 17

medium
Solve 3xβˆ’12=x+4\frac{3x - 1}{2} = x + 4.

Example 18

medium
Solve 3x+2(xβˆ’1)=133x + 2(x - 1) = 13.

Example 19

medium
Solve x2+x3=5\frac{x}{2} + \frac{x}{3} = 5.

Example 20

challenge
Solve for xx: ax+b=ca x + b = c (with a≠0a \ne 0), in terms of aa, bb, cc.

Example 21

challenge
A number doubled, plus 77, equals three times the number minus 22. Find the number.

Example 22

challenge
Solve x2βˆ’xβˆ’34=x+12\frac{x}{2} - \frac{x-3}{4} = \frac{x+1}{2}.

Example 23

easy
Solve 3xβˆ’4=113x - 4 = 11.

Example 24

easy
Solve x5βˆ’2=3\frac{x}{5} - 2 = 3.

Example 25

easy
Solve 5βˆ’x=25 - x = 2.

Example 26

easy
Solve xβˆ’13=4\frac{x-1}{3} = 4.

Example 27

easy
Solve βˆ’2x+7=1-2x + 7 = 1.

Example 28

medium
Solve 3x+5=2(x+7)3x + 5 = 2(x + 7).

Example 29

medium
Solve x3+x6=5\frac{x}{3} + \frac{x}{6} = 5.

Example 30

medium
Solve 5(xβˆ’2)βˆ’3(x+1)=75(x - 2) - 3(x + 1) = 7.

Example 31

medium
Solve 3x+2βˆ’5x=8βˆ’x3x + 2 - 5x = 8 - x.

Example 32

medium
Solve 2(3xβˆ’1)+4=5x+72(3x - 1) + 4 = 5x + 7.

Example 33

medium
Solve 3x4βˆ’12=x2+1\frac{3x}{4} - \frac{1}{2} = \frac{x}{2} + 1.

Example 34

medium
Solve 4(x+1)βˆ’2=3(xβˆ’2)+54(x + 1) - 2 = 3(x - 2) + 5.

Example 35

hard
Solve x+34βˆ’xβˆ’13=16\frac{x + 3}{4} - \frac{x - 1}{3} = \frac{1}{6}.

Example 36

hard
Solve 2xβˆ’15=x+43\frac{2x - 1}{5} = \frac{x + 4}{3}.

Example 37

hard
Two consecutive integers add to βˆ’15-15. Set up and solve.

Example 38

hard
Solve for xx in terms of aa and bb: a(xβˆ’b)=bx+aa(x - b) = bx + a.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

solving linear equationsdistributive propertyexpressions