Two or more equations sharing variables. Solve all of them at once.
Graph
y = 5 - xy = x - 1
The two lines plotted are x + y = 5 (as y = 5 − x) and x − y = 1 (as y = x − 1). Their marked intersection (3, 2) is the unique solution that satisfies both equations.
Your turn
Solve x + y = 7 and x − y = 3 by elimination.
What the graph tells you
- One intersection ⇒ exactly one solution.
- Parallel lines (same slope, different intercept) ⇒ no solution.
- Same line ⇒ infinitely many solutions.
Three methods
- Substitution — solve one for a variable, sub into the other.
- Elimination — add or subtract equations to cancel a variable.
- Matrix — write Ax = b, multiply by A⁻¹.
Try it
x + y = 5, x − y = 1
Add: 2x = 6, x = 3. Then y = 2.