Linear Equations with Decimals Practice Test

Solve decimal equations accurately by aligning place values, clearing decimals, and checking practical models.

Linear Equations with Decimals Practice Test

20 decimal-based linear equation questions with detailed worked explanations.

Instant feedback · Worked explanations

After the test · decimal equation console

Decimal equations are ordinary linear equations with an extra demand: keep place value and precision under control

The algebraic rules do not change when coefficients and constants are decimals. You still preserve equality, simplify both sides, collect variable terms, and isolate the unknown. The main additional decisions are whether to work directly with decimals or first multiply by a power of 10 to convert the equation into an equivalent integer-coefficient form.

Decimal coefficientsSigned valuesPowers of 10Variables on both sides PrecisionRatesPricingSubstitution check

Choose the cleaner representation. An equation such as 0.3x + 1.2 = 4.8 can be solved directly or multiplied by 10 to become 3x + 12 = 48.

0.4

four tenths

0.75

75 hundredths

1.20

same as 1.2

−0.6

sign travels with value

Power-of-ten clearing

Multiplying by 10, 100, or 1000 can remove decimal points from the entire equation at once

Start 0.3x + 1.2 = 4.8

All decimals have one decimal place.

Scale equation 10(0.3x + 1.2) = 10(4.8)

Multiply every term on both sides.

Integer form 3x + 12 = 48

The new equation has the same solution.

Solve 3x = 36 → x = 12

No decimal arithmetic remains.

Solve terminal · 1.5x − 2.4 = 6.6

1.5x − 2.4 = 6.6 Identify the variable term and constant.

1.5x = 9.0 Add 2.4 to both sides.

x = 9.0 ÷ 1.5 Undo multiplication by 1.5.

x = 6 The exact result happens to be an integer.

✓

1.5(6) − 2.4 = 6.6 9.0 − 2.4 = 6.6, so the solution checks.

Equation architectures

The decimal location tells you which operation is likely to simplify the equation first

Decimal coefficient

0.6x + 4 = 10

Remove 4, then divide by 0.6.

Several decimals

0.25x + 1.5 = 3.75

Direct solving works, or multiply by 100.

Variable both sides

1.2x + 3 = 0.7x + 8

Collect x terms before isolating the constant.

Decimal divisor

x/0.5 + 2 = 10

Recognize that dividing by 0.5 doubles the quantity.

Variables on both sides

Decimal coefficients do not change the logic of collecting variable terms

Choose a direction that keeps the remaining coefficient easy to work with.

Original

1.2x + 3 = 0.7x + 8

Subtract 0.7x

0.5x + 3 = 8

Subtract 3

0.5x = 5

Divide by 0.5

x = 10

Precision control

Exact values and rounded approximations should not be mixed without a reason

Exact terminating result

0.8x = 5.6
x = 7

No rounding is needed because the quotient terminates exactly.

Nonterminating result

0.3x = 1
x = 10/3 = 3.333…

Keep 10/3 unless the question explicitly asks for a decimal approximation.

Applied decimal dashboard

Decimal linear equations often model prices, measured rates, or unit costs

Base service fee

$6.50

Cost per unit

$2.75

Number of units

Total bill

$28.50

Equation

6.50 + 2.75x = 28.50

Solution

x = 8

Reasonableness monitor

Estimate the scale of the answer before accepting the exact decimal arithmetic

0.5x = 12 x should be twice 12, so expect about 24.

2.5x = 10 x should be smaller than 10 because the coefficient exceeds 1.

x + 3.8 = 2.1 x must be negative.

−0.4x = 8 x must be negative because a negative times x is positive.

Decimal error diagnostics

Most decimal-equation mistakes are either balance errors or place-value errors

0.4x + 1.2 = 5.2 → 0.4x = 6.4

0.4x = 4.0 → x = 10

Undo +1.2 with subtraction, not addition.

0.25x = 5 → x = 1.25

x = 5 ÷ 0.25 = 20

Dividing by 0.25 is the same as multiplying by 4.

0.3x + 1.2 = 4.8 → ×10 → 3x + 1.2 = 48

3x + 12 = 48

Scaling by 10 applies to every decimal term.

0.3x = 1 → x = 3.3

x = 10/3 ≈ 3.33

Premature rounding changes the exact solution.

Diagnostic dials

Use missed questions to identify which decimal skill is interfering with the algebra

The goal is to separate structural equation solving from numerical precision mistakes.

Place value Decimal addition, subtraction, signs.

Scaling Multiplying the whole equation by 10, 100, or 1000.

Isolation Variables on both sides and decimal coefficients.

Precision Exact answers, rounding, and substitution checks.