Solving Linear Equations Examples: 57 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Solving Linear 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

The process of finding the value of the variable that makes a linear equation true, using inverse operations to isolate the variable on one side of the equals sign. A linear equation has the variable raised only to the first power, producing exactly one solution.

Undo what's done to x by doing the opposite: if x+5, subtract 5.

Read the full concept explanation →

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: Solving a linear equation is preserving equality while isolating the unknown.

Common stuck point: The procedure for solving linear equations is the easy part; the trap is doing an operation to only one side. Asking "Is there an equals sign and a variable value to find?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there an equals sign and a variable value to find?

Worked Examples

Example 1

easy
Solve 3x+7=22.

Answer

x=5

First step

1
Subtract 7 from both sides: 3x=22−7=15.

Full solution

  1. 2
    Divide both sides by 3: x=153=5.
  2. 3
    Check: 3(5)+7=15+7=22 ✓
To solve a linear equation, isolate x by performing inverse operations. Always verify your answer by substituting back into the original equation.

Example 2

medium
Solve 2(x−3)+4=3x−8.

Example 3

medium
Solve 2(3x−4)=5x+6.

Example 4

easy
Solve x5−2=3.

Example 5

medium
Solve 6−2(x+1)=4x−8.

Example 6

medium
A taxi charges $3 plus $2 per mile. If the total fare is $17, how many miles was the ride?

Example 7

hard
A rectangle's length is 3 cm more than twice its width. If the perimeter is 36 cm, find the width.

Example 8

hard
A father is three times as old as his son. In 12 years, he will be twice as old as his son. Find the son's current age.

Practice Problems

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

Example 1

easy
Solve 5x−3=12.

Example 2

hard
Solve 2x+13=x−22.

Example 3

easy
Solve x+5=12.

Example 4

easy
Solve x−8=3.

Example 5

easy
Solve 3x=21.

Example 6

easy
Solve x4=6.

Example 7

easy
Solve 2x+5=11.

Example 8

easy
Solve 5x−3=17.

Example 9

easy
Solve x3+2=7.

Example 10

easy
Solve −x=7.

Example 11

medium
Solve 3(x+4)=21.

Example 12

medium
Solve 2x+7=4x−3.

Example 13

medium
Solve x+34=5.

Example 14

medium
Solve 5−2x=13.

Example 15

medium
Solve 2(x−1)+3=5(x+2)−12.

Example 16

medium
Solve 2x−13=x+24.

Example 17

medium
Three consecutive integers sum to 42. Find them.

Example 18

medium
Sarah is 3 years older than her brother Tom. The sum of their ages is 25. How old is each?

Example 19

medium
Solve 12x+13x=10.

Example 20

challenge
Solve 2x−35−x+12=1.

Example 21

challenge
For what value of k does the equation 3x+7=kx−2 have NO solution?

Example 22

challenge
Anna can paint a room in 4 hours. Ben can paint it in 6 hours. How long if they work together?

Example 23

easy
Solve x+9=16.

Example 24

easy
Solve 7x=56.

Example 25

easy
Solve 4x+9=33.

Example 26

easy
Solve 10−x=4.

Example 27

easy
Solve −3x=18.

Example 28

medium
Solve 5(x−3)=2x+9.

Example 29

medium
Solve 4x+7=2x+23.

Example 30

medium
Solve 23x+5=11.

Example 31

medium
Solve x−45=3.

Example 32

medium
Solve 0.4x+1.2=3.6.

Example 33

medium
Solve 7(2x−1)−3x=4x+14.

Example 34

medium
Solve 3(x+2)−4(x−1)=5.

Example 35

hard
Solve 3x−24+x+13=2.

Example 36

hard
Solve 0.25(8x−4)=1.5x+1.

Example 37

hard
Solve 2x+5=2x+5.

Example 38

hard
Solve 3(x−4)=3x−7.

Example 39

challenge
Solve for x in terms of a and b: a(x−b)=b(x+a), where a≠b.

Example 40

challenge
Train A leaves a station at 60 mph. Two hours later, train B leaves the same station traveling the same direction at 80 mph. How many hours after B's departure does B catch A?

Example 41

easy
Fill in the blank so that the solution is x=5: 3x+□=19. What number goes in the blank?

Example 42

easy
Fill in the blank so that the solution is x=9: x−□=4. What number goes in the blank?

Example 43

medium
Fill in the blank so that the solution is x=3: 4x−□=3x+2. What number goes in the blank?

Example 44

medium
Sam solves x+6=20 like this: "x=20+6, so x=26." Find Sam's mistake and enter the correct value of x.

Example 45

medium
Ava solves 4x−3=2x+5 like this: "4x−2x=5−3, so 2x=2 and x=1." Find Ava's mistake and enter the correct value of x.

Example 46

easy
Solve x−6=9. (Compare with its twin: 6−x=9.)

Example 47

medium
Solve 6−x=9. (Compare with its twin: x−6=9.)

Example 48

easy
A frog starts at a mystery number x on the number line, hops 6 steps to the left, and lands on 10. What number did the frog start on?

Example 49

medium
Mia and Ben collect trading cards. Mia gives away 6 of her cards, Ben finds 6 more cards, and now they have exactly the same number. How many more cards than Ben did Mia have at the start?
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Background Knowledge

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

equationsorder of operations