Systems of Equations
Grades 9–11 Algebra Equations
A system asks: which (x, y) satisfies both equations? On the graph, it's exactly where the two lines cross.
Method 1 — substitution
One equation gives a variable alone; pour it into the other:
y = x + 3
2x + y = 9
Pour the first into the second:
2x + (x + 3) = 9
3x + 3 = 9
3x = 6
x = 2 → y = 2 + 3 = 5
Answer: (2, 5) Check: 2·2 + 5 = 9 ✓
Method 2 — elimination
Add the equations so a variable cancels:
x + y = 8
x − y = 2
───────
2x = 10 ← +y and −y cancel
x = 5 → y = 3
Answer: (5, 3)
Three possible outcomes
- Lines cross once → one solution
- Parallel (same slope, different intercept) → no solution
- Same line twice → infinitely many solutions
Practice
y = x + 1 and y = 7 meet where x = ___
x + y = 6, x − y = −2 → x = ___
y = 2x + 1 and y = 2x + 5 have:
x + y = 10, y = 6 → x = ___
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