Systems of equations on the SAT: lesson and practice
A system of equations is two equations that must both be true. On the SAT, solve it by substitution (replace one variable using the other equation) or elimination (add or subtract the equations to cancel a variable). Always check what the question asks for: x, y, or an expression like x + y.
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Two methods
| Method | Use it when | Example |
|---|---|---|
| Substitution | One equation already says y = … or x = … | y = 2x + 1 and 3x + y = 16 → 3x + 2x + 1 = 16 → x = 3 |
| Elimination | Coefficients match or are easy to match | x + y = 10 and x − y = 4 → add: 2x = 14 → x = 7 |
How many solutions?
| Lines | Solutions |
|---|---|
| Different slopes | Exactly one |
| Same slope, different intercepts (parallel) | None |
| Same line | Infinitely many |
Word problems
Write one equation for the count and one for the total. 40 tickets, Rs 400 adult and Rs 200 student, Rs 12,000 collected: a + s = 40 and 400a + 200s = 12,000.
Common mistakes
Sample questions with answers
1. If x + y = 10 and x − y = 4, what is y?
- 7
- 3 (correct)
- 14
- 6
Add to get 2x = 14, so x = 7 and y = 3. Choice A gives x instead.
2. For what value of k does 2x + ky = 5 and 4x + 6y = 7 have no solution?
- 6
- 3 (correct)
- −3
- 12
The second equation is twice the first on the left if k = 3, and 7 is not 2 × 5.
3. If 3x + 2y = 17 and x + 4y = 19, what is 4x + 6y?
- 2
- 36 (correct)
- 8
- 17
Add the equations: 4x + 6y = 17 + 19 = 36.
Frequently asked questions
How many systems of equations questions are on the SAT?
Systems are part of Algebra, which makes up about 35% of SAT Math, roughly 13 to 15 questions. Several of those usually involve systems.
Is substitution or elimination better for the SAT?
Use substitution when one variable is already isolated and elimination when the coefficients line up. Desmos is a quick way to check either.
When does a system have no solution?
When the two lines have the same slope but different intercepts, so they are parallel and never meet.
Can I use Desmos for systems of equations?
Yes. Type both equations into the built-in Desmos calculator and click the point where the lines intersect.