3.1 Systems of equations (substitution & elimination)
Key Takeaways
- A 2×2 system's solution is the ordered pair (x, y) that satisfies both equations at once — the point where the two lines cross.
- Substitution isolates one variable and plugs it into the other equation; use it when a variable is already alone or easy to isolate.
- Elimination adds or subtracts the equations (after scaling if needed) to cancel a variable; both methods give the same solution.
- A false result like 0 = 8 means no solution (parallel lines); a true result like 0 = 0 means infinitely many solutions (same line).
- For word problems, define variables, write one equation for the count and one for the total, solve, then check both equations.
Two equations, one point
A system of equations is a set of two or more equations that share the same variables. On the PERT you will almost always face a 2×2 linear system: two equations, each containing an x and a y. Solving the system means finding the ordered pair (x, y) that makes both equations true at the same time. Graphically, each linear equation is a line, and the solution is the single point where the two lines cross.
Two reliable algebra methods appear on the test: substitution and elimination. Below we solve the same system both ways so you can confirm that they produce an identical answer. Choosing between them is a matter of convenience — a system that looks awkward for one method is often quick for the other.
Our system:
- Equation 1:
2x + y = 11 - Equation 2:
x - y = 1
Method 1 — Substitution
Substitution works best when one variable is already alone, or is easy to isolate because its coefficient is 1.
- Isolate a variable. Equation 2 rearranges to x = y + 1.
- Substitute that expression into Equation 1:
2(y + 1) + y = 11. - Distribute and combine like terms:
2y + 2 + y = 11, so3y + 2 = 11. - Solve the one-variable equation:
3y = 9, so y = 3. - Back-substitute into
x = y + 1:x = 3 + 1 = 4.
The solution is (4, 3). Always check it in both originals: 2(4) + 3 = 11 ✓ and 4 - 3 = 1 ✓. That check is the single best guard against a sign slip.
Method 2 — Elimination
Elimination works best when adding or subtracting the equations cancels a variable outright.
Notice Equation 1 has +y and Equation 2 has −y. Add the two equations term by term:
(2x + y) + (x - y) = 11 + 1 → 3x = 12 → x = 4.
Substitute x = 4 back into Equation 2: 4 - y = 1, so y = 3. The answer is again (4, 3) — the two methods must agree, and they do.
When a variable will not cancel on its own, multiply one or both equations by a constant first so that the coefficients of one variable become opposites. For instance, to eliminate x from 2x + y = 11 and x - y = 1, multiply Equation 2 by −2 to get -2x + 2y = -2, then add it to Equation 1: 3y = 9, giving y = 3 once more.
When neither variable cancels
Many systems need scaling on both equations. Consider 3x + 2y = 16 and 2x + 5y = 18. To eliminate x, multiply the first equation by 2 and the second by 3 so both x-coefficients become 6: 6x + 4y = 32 and 6x + 15y = 54. Subtract the first new equation from the second: 11y = 22, so y = 2. Substitute into 3x + 2(2) = 16 to get 3x = 12, so x = 4. The solution (4, 2) checks in both originals. The lesson: line up one variable's coefficients as opposites (so you add) or as equals (so you subtract), then knock that variable out.
How many solutions? Three cases
Not every system crosses at one point. The PERT expects you to recognize all three outcomes, which correspond to the slopes and intercepts of the two lines.
| Case | What happens algebraically | Lines | Solutions |
|---|---|---|---|
| One solution | Variables solve to unique numbers | Cross once (different slopes) | Exactly one (x, y) |
| No solution | You reach a false statement like 0 = 8 | Parallel (same slope, different intercept) | None |
| Infinitely many | You reach a true statement like 0 = 0 | Same line (identical) | Every point on the line |
Example of no solution: y = 2x + 1 and y = 2x - 3. Setting them equal gives 2x + 1 = 2x - 3, then 1 = -3, which is false — the lines are parallel and never meet. Example of infinitely many: y = 2x + 1 and 2y = 4x + 2. The second equation is just the first doubled, so every point on the line satisfies both, and the algebra collapses to 0 = 0.
A word problem
Systems shine on real-world questions with two unknowns. Suppose a school sells 200 tickets to a play. Adult tickets cost $9 and child tickets cost $6, and the total collected is $1,530. How many of each were sold?
Let a = adult tickets and c = child tickets. Translate the two facts into equations:
- Count:
a + c = 200 - Money:
9a + 6c = 1530
Use substitution: c = 200 - a. Then 9a + 6(200 - a) = 1530 → 9a + 1200 - 6a = 1530 → 3a = 330 → a = 110. So c = 200 - 110 = 90. Check the money: 9(110) + 6(90) = 990 + 540 = 1530 ✓. The school sold 110 adult and 90 child tickets.
Whichever method you choose, the workflow is the same: translate the words into two equations, solve for one variable, substitute back for the other, and verify the pair in both original equations before you bubble in an answer.
Solve the system x + y = 10 and x - y = 4 by elimination. What is the ordered pair (x, y)?
How many solutions does the system y = 3x + 2 and y = 3x - 5 have?
A theater sells 50 tickets total. Adult tickets are $12 and student tickets are $8, and the total collected is $520. How many adult tickets were sold?