Solving Linear Equations Examples in Math

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

Subtract 7 from both sides:

3x = 22 - 7 = 15

Full solution

2
Divide both sides by 3: $x = \frac{15}{3} = 5$ .
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

\frac{x}{5} - 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

\frac{2x - 1}{3} = \frac{x + 2}{4}

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

\frac{1}{2}x + \frac{1}{3}x = 10

Example 20

challenge

Solve

\frac{2x - 3}{5} - \frac{x + 1}{2} = 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

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 \neq 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?

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Related Concepts

Equations Order of Operations Linear Functions Systems of Equations

Background Knowledge

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

equationsorder of operations