What you will be able to do
- Work each problem without looking at the answer.
- Write the algebraic step that justifies each change.
- Check the result in the original relationship.
- Use the error notes to decide what to practise next.
How to use this worksheet
Choose a method based on the equations. Show how one variable is eliminated or substituted, then check the final ordered pair in both originals.
- Select an efficient method
- Maintain equality when multiplying equations
- Interpret one, no or infinitely many solutions
- Check an ordered pair in both equations
For a clean student copy, print this page and cover the worked solutions or select the problem pages in your browser’s print preview.
Practice problems
Problem 1
y = 2x + 1 and y = −x + 7
Problem 2
2x + y = 11 and x − y = 1
Problem 3
3x + 2y = 4 and 6x + 4y = 8
Problem 4
2x − y = 5 and 4x − 2y = 3
Problem 5
3x + 4y = 18 and 5x − 4y = 14
Answer key with worked solutions
Problem 1
y = 2x + 1 and y = −x + 7
- 1.Set expressions equal: 2x + 1 = −x + 7.
- 2.3x = 6, so x = 2.
- 3.y = 2(2) + 1 = 5.
Answer: (2, 5)
Check: 5 = −2 + 7.
Problem 2
2x + y = 11 and x − y = 1
- 1.Add equations: 3x = 12, so x = 4.
- 2.Substitute: 4 − y = 1, so y = 3.
Answer: (4, 3)
Check: 2(4) + 3 = 11.
Problem 3
3x + 2y = 4 and 6x + 4y = 8
- 1.The second equation is twice the first.
- 2.They represent the same line.
Answer: Infinitely many solutions
Check: Every solution of the first satisfies the second.
Problem 4
2x − y = 5 and 4x − 2y = 3
- 1.Doubling the first gives 4x − 2y = 10.
- 2.This contradicts 4x − 2y = 3.
Answer: No solution
Check: The lines are parallel and distinct.
Problem 5
3x + 4y = 18 and 5x − 4y = 14
- 1.Add equations: 8x = 32, so x = 4.
- 2.Substitute: 12 + 4y = 18, so y = 1.5.
Answer: (4, 1.5)
Check: 20 − 6 = 14.
Use mistakes to choose the next step
Learn substitution and elimination step by stepReview graphing lines
| If this happened | Likely issue | Next practice move |
|---|---|---|
| Solution satisfies only one equation | Incomplete check or arithmetic error | Substitute into both originals |
| Variables do not cancel | Multipliers were not applied to every term | Rewrite the full scaled equation |
| 0 = 0 or 0 = nonzero appears | Special solution type | Interpret identity as infinite and contradiction as none |
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Useful answers
Frequently asked questions
How do I choose between substitution and elimination?
Use substitution when a variable is already isolated or has coefficient 1. Use elimination when coefficients already match or are easy to match.
What does no solution look like?
Algebra leaves a contradiction such as 0 = 7; graphically the lines are parallel.
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Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.