Equations With Variables on Both Sides Worksheet

How it works

Twelve equations like 9x + 9 = 7x + 25 open this free page, three work lines under each. Sheet type adds negatives, one bracket, brackets on both sides, fraction answers, and equations with no solution or every solution.

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Collect the letter on one side

Take the smaller x term from both sides so the letter stays positive: 9x + 9 = 7x + 25 becomes 2x + 9 = 25, then 2x = 16 and x = 8. The key prints the collected line, 2x = 16, then 16 ÷ 2 = 8, then both sides checked: 81 and 81.

When the letters cancel

If the x terms match, they vanish. 3x − 2 = 3x + 12 leaves −2 = 12, which is false, so there is no solution. 2(x + 1) = 2x + 2 leaves 2 = 2, true for every number. Common Core 8.EE.C.7a asks for all three outcomes.

Print Fraction answers after Negative numbers: 6x − 11 = 3x − 1 gives x = 10/3, and pupils who expect whole numbers stop checking.
On No solution and all real numbers, ask pupils to write the last line, such as −2 ≠ 12, before the words.

Questions and answers

Which side should the x go on?

Either works, but moving the smaller x term keeps the coefficient positive. In 4y + 17 = 8y − 15, take 4y away: 17 = 4y − 15, so 32 = 4y and y = 8. The key's −4y = −32 gives the same answer.

Are the answers always whole numbers?

On eight of the ten sheet types, yes. Fraction answers and Fraction answers with a bracket give answers such as 10/3 and 5/2, always in lowest terms.

How do I print it as a PDF?

Press Print and choose Save as PDF as the printer. The answer key follows the sheet on its own page; untick Answer key to leave it out.

All 10 sheet types

Whole-number answers 2

Negative numbers: 4y + 17 = 8y − 15 4y + 17 = 8y − 15 gives −4y = −32, so y = 8. Coefficients from −9 to 9 and answers from −12 to 12.
Positive numbers: 9x + 9 = 7x + 25 9x + 9 = 7x + 25: take 7x from both sides to get 2x + 9 = 25, so 2x = 16 and x = 8. Every number in the equations is positive.

With brackets 3

All three forms mixed ax + b = cx + d, a(x + b) = cx + d and a(x + b) = c(x + d) shuffled, whole-number answers from −12 to 12.
Brackets on both sides: 6(x + 5) = 2(x − 1) 6(x + 5) = 2(x − 1): expand to 6x + 30 = 2x − 2, collect to 4x = −32, so x = −8.
One bracket: 5(x − 6) = 7x − 8 5(x − 6) = 7x − 8: expand to 5x − 30 = 7x − 8, collect to −2x = 22, so x = −11.

Fraction answers and special cases 3

Fraction answers with a bracket 2(x + 5) = 8x − 5: expand to 2x + 10 = 8x − 5, collect −6x = −15, so x = 5/2. Every answer is a fraction in lowest terms.
Fraction answers: 6x − 11 = 3x − 1 6x − 11 = 3x − 1 gives 3x = 10, so x = 10/3. About one answer in four is whole; the rest are fractions in lowest terms.
No solution and all real numbers 3x − 2 = 3x + 12 has no solution, because −2 ≠ 12; 2(x + 1) = 2x + 2 is true for all real numbers. About one problem in four is one of these; the rest have one answer.

Checking and quick sheets 2

Quick sheet: 20 across, no work space 20 short equations such as 6x + 6 = x + 26 in rows, for mental work or a warm-up; the key gives each answer.
Solve, then check both sides A work line and a Check line under each equation, some with a bracket. For 4x + 3 = 5x − 2 the key writes 4 × 5 + 3 = 23 and 5 × 5 − 2 = 23.

Notes

Sheets, puzzles and cards made here are practice material. They are not an assessment, they follow no single curriculum, and they do not replace a specialist's advice about a child's handwriting or learning.

More notes about this page

Facts on this page were checked against standard reference sources. If you spot a mistake, please tell us through the contact page.

How this list was compiled: Our own list of sheet types, written on 2026-09-24. Every equation is made backwards from its answer in exact fractions, and a second program reads each printed equation again and solves it, 10,000 problems per sheet type. Common Core 8.EE.C.7a (linear equations in one variable with one solution, infinitely many solutions, or no solutions, shown by transforming the equation until x = a, a = a or a = b results) and 8.EE.C.7b (solve linear equations with rational number coefficients, including those that need the distributive property and like terms collected), read on 2026-09-24 at https://tasks.illustrativemathematics.org/content-standards/8/EE/C/7 . The standard is a checklist for teachers; its authors have not reviewed these sheets. Licence: own work

Equations With Variables on Both Sides Worksheet for Grade 8