Two equations, two unknowns
A system of linear equations is a set of equations that must be true at the same time:
- x + y = 10
- x − y = 2
The solution is the pair (x, y) that works in both: (6, 4).
Method 1: Graphing
Graph both lines. The point of intersection is the solution.
| Graph | Number of solutions | Description |
|---|---|---|
| Lines cross | one | consistent, independent |
| Lines parallel | none | inconsistent |
| Same line | infinitely many | consistent, dependent |
Graphing is visual but can be imprecise when the answer isn’t a whole number.
Method 2: Substitution
Best when one variable is already isolated (or easy to isolate).
Example:
- y = 2x − 1
- 3x + y = 14
- Substitute y = 2x − 1 into the second: 3x + (2x − 1) = 14
- 5x − 1 = 14 → x = 3
- Back-substitute: y = 2(3) − 1 = 5
Solution: (3, 5)
Method 3: Elimination
Best when the equations are in standard form (Ax + By = C).
Example:
- 2x + 3y = 12
- 4x − 3y = 6
- Add the equations (the y terms cancel): 6x = 18 → x = 3
- Substitute: 2(3) + 3y = 12 → y = 2
Solution: (3, 2)
When coefficients don’t match, multiply first.
- 3x + 2y = 16
- 2x + 5y = 18
Multiply the first by 2 and the second by 3: 6x + 4y = 32 and 6x + 15y = 54. Subtract: 11y = 22 → y = 2. Then 3x + 4 = 16 → x = 4. Solution: (4, 2).
Always check
Substitute into both original equations.
(4, 2): 3(4) + 2(2) = 16 ✓; 2(4) + 5(2) = 18 ✓
Word problem
Adult tickets cost $12 and student tickets cost $8. 150 tickets were sold for $1,480. How many of each?
- a + s = 150
- 12a + 8s = 1,480
From the first: s = 150 − a. Substitute: 12a + 8(150 − a) = 1,480 → 4a = 280 → a = 70, s = 80.
70 adult, 80 student tickets.
Worked examples
Example 1: x + y = 9, x − y = 3 → add: 2x = 12 → (6, 3).
Example 2: y = x + 4, 2x + y = 13 → 3x + 4 = 13 → (3, 7).
Example 3: 2x + y = 5, 4x + 2y = 7 → doubling the first gives 4x + 2y = 10 ≠ 7 → no solution (parallel lines).
Common mistakes
- Substituting back into the same equation you rearranged (it always works, so it doesn’t check anything).
- Sign errors when subtracting equations.
- Multiplying only one side of an equation.
- Giving only x without y.
Try it yourself
- x + y = 12, x − y = 4
- y = 3x, x + y = 20
- 3x + 2y = 7, x − 2y = 5
- A farm has chickens and goats: 30 heads, 84 legs. How many of each?
Answers: 1) (8, 4) 2) (5, 15) 3) (3, −1) 4) 18 chickens, 12 goats
Quick recap
- A system of linear equations is solved by graphing, substitution or elimination.
- The solution is the point of intersection.
- Systems can have one, none or infinitely many solutions.