Simultaneous Equations Worksheets

What this worksheet covers

This printout presents pairs of equations and asks for the values that satisfy both conditions together. Its separate marking page can include each elimination or substitution step. Generate another copy to get different coefficients and constants.

What a sheet of this kind contains:

• solve x + y = 13 and x − y = 3
• solve 3x + y = 17 and x − y = 1
• solve y = 2x + 1 and 4x + y = 13
• find two numbers that add to 30 and differ by 8
• solve x + 2y = 11 and x − 2y = 1
• check both answers in both equations
• solve 2x + y = 12 and x − y = 3

Choosing a level

The distance from ready-to-cancel coefficients sets the climb. Warm-Up lets opposite y terms disappear by addition; Basics uses matching signs that call for subtraction. Practice needs one equation scaled first. Steady offers a variable already isolated for substitution. Challenge translates a short word problem, and Mastery combines both methods, sometimes with a negative or fractional result.

Set any whole-number count from 3 through 15. Same difficulty repeats the selected stage, while Mixed difficulties moves up the six-stage sequence from there. Use the latter for a recap after both elimination and substitution have been taught.

Choose Write-in when learners need to show the elimination or substitution they used. A–D is quicker for a short selection check, but it cannot show where an arithmetic or sign error entered the working.

A worked example

The first pair cancels a variable; the second replaces one with an expression.

Solve x + y = 13 and x − y = 3. The y terms are opposites, so add the equations: 2x = 16, giving x = 8. Put that back into the opening line: 8 + y = 13, so y = 5. Verify: 8 + 5 = 13 and 8 − 5 = 3.

Solve y = 2x + 1 and 4x + y = 13. The top equation has y on its own, so swap it in: 4x + 2x + 1 = 13, which is 6x = 12, so x = 2. Then y = 2 × 2 + 1 = 5. Verify: 4 × 2 + 5 = 13.

Marking with the answer key

Watch the operation applied to the right-hand sides as carefully as the variables.

• 2x = 10 after adding x + y = 13 and x − y = 3. The y terms were cancelled, and then the 13 and the 3 were taken away from each other as well. Both right-hand sides are added: 13 + 3 = 16, so x = 8 and y = 5.
• x = 8 offered as the whole answer. One letter is only part of a solution. Substituting 8 into x + y = 13 gives y = 5, and the verification covers both equations.
• y = 2x − 1 written in place of y = 2x + 1. The sign changed while the expression was being copied. With 4x + y = 13 the wrong version gives 6x = 14, so x is not even a whole number: x = 2, y = 5 is the pair to write down.
• 15 and 15 for two numbers that add to 30 and differ by 8. Halving uses only the opening fact. The pair is 19 and 11, which adds to 30 and shows a difference of 8.

Printing and using it

Print one solved pair at the top as a method prompt. Showing the right-hand sides being added or subtracted gives pupils a model to consult while they set up the next pair.

Standard leaves a line under each equation for the scaling step. Large-print suits a younger group still learning where the signs belong, and Compact fits twelve short pairs onto one side for a starter.

Turn on the Name / Date / Class line for collected work. Dyslexia-friendly type gives the minus signs more room in expressions such as −y and −5. Stretch raises every third item by a rung on a same-difficulty sheet. Rename the page "Find both values" if that makes the task clearer to the group.

For a later practice session, cover the model pair and ask learners to reconstruct the method from memory before checking the separate key.

Keep going

Work the same pairs with instant marking at the simultaneous equations practice ladder, read the simultaneous equations lesson for the graphs behind the algebra, or print linear equations worksheets, where one equation at a time is solved and rearranged.

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