Linear Equations with Fractions Practice Test

Clear denominators, preserve equality, and solve fractional linear equations with confidence.

Linear Equations with Fractions Practice Test

20 fraction-based linear equations with instant feedback and step-by-step solutions.

Instant feedback · Worked explanations

After the test · fraction equation notebook

Fractions make linear equations look harder than they are because two jobs happen at once: fraction arithmetic and equation solving

The most reliable strategy is to separate those jobs. First decide whether denominators can be cleared cleanly. Then solve the resulting linear equation using ordinary balance operations. When fractions contain variable expressions, also keep track of values that would make a denominator zero.

LCDClearing denominatorsFractional coefficients ReciprocalsVariable denominatorsRestrictionsSolution checking

A useful habit: write the LCD in the margin before multiplying anything. This turns a cluttered fraction equation into a controlled algebra step.

Equation x/3 + 5/6 = 7/2 denominators: 3, 6, 2

LCD 6 smallest common multiple

Multiply 6(x/3) + 6(5/6) = 6(7/2) every term receives 6

Clean equation 2x + 5 = 21 fractions removed

Solve 2x = 16 → x = 8 ordinary linear equation

LCD clearing strip

Clearing denominators is one equation operation, not several unrelated cancellations

Inventory x/4 + x/6 = 5

Denominators are 4 and 6.

Choose LCD LCD = 12

12 is divisible by both denominators.

Multiply whole equation 3x + 2x = 60

The right side is multiplied by 12 too.

Finish 5x = 60 → x = 12

Now solve normally.

Worked page · (2x − 1)/3 + (x + 2)/2 = 7

LCD = 6 Both denominators 3 and 2 divide 6.

2(2x − 1) + 3(x + 2) = 42 Multiplying the full equation by 6 clears both denominators.

4x − 2 + 3x + 6 = 42 Distribute after the fractions are gone.

7x + 4 = 42 Combine like terms.

7x = 38 → x = 38/7 An exact fractional answer is completely valid.

Where the fraction sits matters

Different fraction structures suggest different first moves

Fractional coefficient

(3/5)x + 2 = 11

Remove 2, then multiply by 5/3.

Several numeric denominators

x/3 + x/4 = 7

Clear denominators with LCD 12.

Grouped numerator

(2x + 5)/4 = 9

Multiply by 4 before separating 2x + 5.

Variable denominator

6/x = 3

Record x ≠ 0, then clear the denominator.

Reciprocal checkpoint

A fractional coefficient is undone by its reciprocal after the variable term is isolated

For (4/7)x − 3 = 9, remove the constant first. Then undo multiplication by 4/7.

(4/7)x − 3 = 9
→ (4/7)x = 12

→

× 7/4
→ x = 21

Restriction notes

If a variable appears in a denominator, record excluded values before transforming the equation

Example

6/x = 3 The original denominator means x ≠ 0.

Clear denominator

6 = 3x → x = 2 The candidate solution does not violate x ≠ 0, so it is valid.

General habit

Write denominator ≠ 0 restrictions before cancellation or multiplication. This keeps domain information visible after the expression becomes simpler.

Solution-set stamps

Fractions do not change the three possible outcomes of a linear equation

x/2 + 3 = 8
→ x/2 = 5
→ x = 10

The variable remains and one value satisfies the equation.

x/3 + 2 = x/3 + 5
→ 2 = 5

The variable cancels and a false statement remains.

(x + 3)/2 = x/2 + 3/2
→ x + 3 = x + 3

The equation becomes an identity, so every allowed value works.

Fraction model · recipe batch

A fraction equation often represents sharing, rates, or a part of a total

Total ingredientx cups

Used in first batchx/3

Used in second batchx/4

Total used7 cups

Equationx/3 + x/4 = 7

Solutionx = 12

Margin corrections

The most common fraction-equation mistakes happen when only part of the equation receives an operation

x/3 + 2 = 5 → x + 2 = 15 Multiply every term by 3: x + 6 = 15

The constant 2 must also be multiplied by 3.

(2/5)x = 8 → x = 16/5 Multiply by 5/2: x = 20

Undo a fractional coefficient with its reciprocal.

x/4 + x/6 = 5 → 6x + 4x = 5 Multiply by 12: 3x + 2x = 60

The right side must receive the LCD multiplier too.

6/x = 3 → x = 0 is allowed Original restriction: x ≠ 0

A denominator cannot be zero, even if later algebra hides the original denominator.

Score ruler

Group missed questions by the fraction skill that caused the breakdown

This separates ordinary linear-equation errors from denominator and fraction-arithmetic errors.

LCD choiceThe clearing multiplier was inefficient or incorrect.

Whole-equation multiplicationOne term was skipped when denominators were cleared.

Reciprocal / arithmeticThe equation structure was correct, but fraction operations failed.

Restrictions / checkingAn excluded value or substitution check was missed.