Systems of equations for the LCAT: lesson and practice
A system of equations is two equations that must both be true. On the LCAT, 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
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.
Is systems of equations tested on the LCAT?
Yes. It falls under Algebra, one of the content domains LUMS lists for the LCAT in its official sample questions.
Can I practice this without a calculator?
Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.