Linear Equations Practice Test

Build a broad foundation in solving, checking, modeling, and rearranging linear equations.

Linear Equations Practice Test

20 mixed linear equation questions with instant feedback and worked solutions.

Instant feedback · Worked explanations

After the test · linear equation solveboard

Solving a linear equation is a sequence of equivalent equations that preserves the same solution

The goal is to isolate the unknown, but the deeper rule is equality preservation. Every legal step must produce an equation with the same solution set. That is why inverse operations, distribution, collecting terms, clearing fractions, and moving variable terms all work only when both sides remain balanced.

Inverse operationsVariables on both sidesParenthesesFractions ProportionsSpecial casesFormula rearrangementModels

Equality balance

Whatever operation changes one side must also change the other side

Left side3x + 5

Right side20

Subtract 5 from both sides: 3x = 15. Then divide both sides by 3: x = 5.

First-step selector

The best first move depends on the structure of the equation

Parentheses?Distribute or simplify grouping first.

Like terms?Collect them on each side before moving terms.

Fractions?Consider clearing denominators with the LCD.

Variable both sides?Move variable terms to one side first.

Variables on both sides

Choose the side that keeps the variable coefficient convenient

5x + 7 = 2x + 22

3x + 7 = 22

3x = 15

x = 5

Subtracting 2x from both sides keeps the remaining coefficient positive. Subtracting 5x would also be legal, but it creates −3x and usually adds unnecessary sign handling.

Several solution paths may be correct. Prefer the path with fewer negatives and smaller coefficients.

Parentheses solve path

Simplify the equation before isolating the variable

Original3(x − 2) + 4 = 19

Distribute3x − 6 + 4 = 19

Collect3x − 2 = 19

Isolatex = 7

Fraction cleaner

Clearing denominators can turn a fraction equation into an ordinary linear equation

Before clearing

x/3 + 1/2 = 5/2

Multiply every term by 6

2x + 3 = 15
2x = 12
x = 6

The LCD multiplies every term on both sides, not just the terms containing fractions you want to remove.

Special solution sets

If the variable disappears, inspect the statement that remains

No solution

3(x + 2) = 3x + 10
3x + 6 = 3x + 10
6 = 10

False statement → no solution

Infinitely many solutions

4(x − 1) = 4x − 4
4x − 4 = 4x − 4
−4 = −4

True identity → all real x

Proportion crossroad

Cross multiplication is shorthand for clearing denominators in a proportion

Proportionx/8 = 15/20

→

Equivalent equation20x = 120
x = 6

The cross products are equal because multiplying both sides by both denominators removes the fractions.

A = ½h(b₁ + b₂)
↓ solve for h
2A = h(b₁ + b₂)
↓
h = 2A/(b₁ + b₂)

Rearranging formulas

Literal equations use the same balance rules even when several letters appear

Treat the variable you want as the unknown and treat the other letters as known quantities.

Start: A = ½h(b₁ + b₂)

Multiply both sides by 2.

Divide by b₁ + b₂.

Result: h = 2A/(b₁ + b₂)

Equation models

A linear model turns a verbal condition into one equality to solve

Consecutive integers

n + (n + 1) + (n + 2) = 48
3n + 3 = 48
n = 15

The integers are 15, 16, and 17.

Pricing

12 + 4x = 40
4x = 28
x = 7

A $12 fixed fee plus $4 per unit reaches $40 at 7 units.

Substitution verification

The original equation is the final judge of a proposed solution

Original5x + 7 = 2x + 22

→

Substitute x = 55(5) + 7 = 2(5) + 22
32 = 32 ✓

Checking the original equation catches sign errors, arithmetic slips, and mistakes introduced during distribution.

Linear-equation error audit

Four shortcuts that break equality or misclassify the result

2x + 5 = 17 → 2x = 22 2x + 5 = 17 → 2x = 12

Subtracting 5 from both sides gives 17 − 5, not 17 + 5.

4(x − 3) = 20 → 4x − 3 = 20 4x − 12 = 20

The factor 4 multiplies both terms inside the parentheses.

3x + 4 = 3x + 9 → x = 5 4 = 9 → no solution

When the variable cancels and a false statement remains, no value of x works.

x/4 + 2 = 5 → x + 2 = 20 Multiply every term by 4: x + 8 = 20

Clearing denominators affects every term on both sides.

Use the score diagnostically

A missed linear equation usually points to a specific stage of the solve path

Classify the error before repeating the test so your next attempt targets the actual weak point.

SimplificationDistribution, fractions, or like terms went wrong first.

Balance operationsAn inverse operation was not applied equally to both sides.

Solution classificationOne, none, or infinitely many solutions was misread.

Modeling & checkingThe equation did not match the context or the result was not verified.