Systems of Equations Practice Problems (Two Variables)

How it works

Elimination: add or subtract the equations so one variable cancels. From x + y = 8 and x − y = 2, adding gives 2x = 10.

Substitution: solve one equation for a variable and plug it into the other. This is best when one coefficient is already 1.

Look at what is asked. If a problem wants x + y, or y + z, try subtracting whole equations instead of solving for everything. Two key identities:

x² + y² = (x + y)² − 2xy x² − y² = (x + y)(x − y)

These problems come from Mathyessy's question bank. Try each one before opening the solution.

If x + y = 8 and x − y = 2, what is x?
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Answer: 5
Add the equations: 2x = 10, so x = 5 (and y = 3).
If x + y + z = 15 and 2x + y + z = 18, what is y + z?
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Answer: 12
Subtract the first equation from the second: x = 3. Then y + z = 15 − 3 = 12. Notice you never need y or z separately.
8 apples and 9 bananas cost 139 cents; 8 apples and 10 bananas cost 150. What does one banana cost?
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Answer: 11 cents
The second purchase is the first plus exactly one more banana, so a banana costs 150 − 139 = 11 cents.
If x + y = 10 and xy = 16, what is x² + y²?
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Answer: 68
Use x² + y² = (x + y)² − 2xy = 100 − 32 = 68. (The numbers are 2 and 8: 4 + 64 = 68.)

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