What you will be able to do
- Substitution is efficient when a variable is isolated.
- Elimination is efficient when coefficients match or can be matched.
- A solution must satisfy both equations.
- Contradictions mean no solution; identities mean infinitely many.
A system asks for a shared solution
Each equation describes a set of points. The solution to the system is the point or points that satisfy both. For two nonparallel linear equations, that is their single intersection.
Algebraic methods preserve the same meaning while avoiding an imprecise graph reading.
Substitution example
Worked example 1
y = 2x + 3 and 3x + y = 18.
- 1.Substitute 2x + 3 for y: 3x + 2x + 3 = 18.
- 2.5x = 15, so x = 3.
- 3.y = 2(3) + 3 = 9.
Answer: (3, 9)
Check: 3(3) + 9 = 18.
Elimination example
Worked example 1
2x + 3y = 13 and 4x − 3y = 11.
- 1.Add equations to eliminate y: 6x = 24.
- 2.x = 4.
- 3.Substitute: 8 + 3y = 13, so y = 5/3.
Answer: (4, 5/3)
Check: 16 − 5 = 11.
Choose the method from the structure
| Structure | Efficient choice | Reason |
|---|---|---|
| One variable already isolated | Substitution | Replacement is immediate |
| Opposite coefficients appear | Elimination | Adding cancels a variable |
| A coefficient of 1 appears | Often substitution | Isolation is simple |
| Small multiples create opposites | Elimination | Scaling is controlled |
| Need visual estimate or interpretation | Graphing | Intersection meaning is visible |
Use a consistent solution routine
Complete the systems worksheetReview linear graphing
- 1Choose and state the method.
- 2Solve one equation for one variable or align coefficients.
- 3Find the first variable.
- 4Substitute back to find the second.
- 5Write an ordered pair.
- 6Check in both original equations and interpret in context.
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Useful answers
Frequently asked questions
Can substitution and elimination give different answers?
No, if both are performed correctly. They solve the same system.
What if both variables cancel?
A true identity indicates infinitely many solutions; a contradiction indicates no solution.
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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.