What are simultaneous equations?
Simultaneous equations are two equations in two unknowns that must be true at the same time. The solution is the single pair (x, y) that fits both — the point where the two lines cross on a graph.
Key idea
Elimination: add or subtract the equations to remove one letter. Substitution: make one letter the subject, then substitute into the other equation.
Elimination method
Worked example
Solve x + y = 7 and x − y = 3.
Add the two equations (the y terms cancel):
2x = 10, so x = 5.
Substitute into x + y = 7: 5 + y = 7, so y = 2.
Check in the second equation: 5 − 2 = 3. ✓
Solution: x = 5, y = 2.
Substitution and word problems
When one equation gives a letter directly (e.g. y = x − 1), substitution is quickest. Many real problems become simultaneous equations: form two equations from the two pieces of information, then solve.
Remember
- Two unknowns need two equations.
- Always find both x and y, and check in both equations.
- Parallel lines never meet — no solution.