The big idea: A pair of simultaneous equations is solved by finding the point that satisfies both at once. On a graph, that is the intersection point.
| Graph picture | Meaning |
|---|---|
| One intersection | one solution |
| No intersection | no solution |
| Same line | infinitely many solutions |
Coordinates matter: The full solution is usually the coordinate pair, not just one number.
Worked example
Two lines intersect at (3, 7). What is the solution of the simultaneous equations?
Step by step
- The point on both graphs is (3, 7).
Final answer
x = 3 and y = 7, so the solution is (3, 7).
Do not report only x: Unless the question specifically asks for just one coordinate, the graphical solution is the intersection point.
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| Situation | Graph behaviour | Conclusion |
|---|---|---|
| Two non-parallel lines | meet once | one solution |
| Parallel distinct lines | never meet | no solution |
| Same line twice | overlap completely | infinitely many solutions |
Say why: If asked to explain, connect the number of solutions directly to how many times the graphs intersect.
Context example
Two cost models intersect at (5, 40). What does this mean?
Step by step
- At x = 5, both models give y = 40.
- So both options cost the same at 5 units / 5 days / 5 items, depending on context.
Final answer
The two models are equal when x = 5, at a value of 40.
Interpret both coordinates: In context questions, explain what x and y each mean in the real situation.