Intersection = the shared point: Two non-parallel lines cross at exactly one point. At that point, both equations are satisfied simultaneously. You solve for x, then substitute to find y.
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Worked example — substitution
Find the intersection of y = 2x − 1 and y = −x + 8.
Step by step
- Set equal (both equal y).
- Solve for x.
- Substitute into either equation.
Final answer
Intersection at (3, 5).
When substitution is awkward, eliminate: If lines are given in ax + by = c form, it is often easier to eliminate one variable by adding or subtracting the equation.
Worked example — elimination
Find the intersection of 2x + 3y = 12 and x − 3y = 0.
Step by step
- Add both equations to eliminate y.
- Substitute x = 4 into x − 3y = 0.
Final answer
Intersection at (4, 4/3).
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| Case | Description | Number of solutions |
|---|---|---|
| Intersecting lines | Different gradients | Exactly one (x, y) point |
| Parallel lines | Same gradient, different y-intercept | No solution |
| Coincident lines | Same gradient AND same y-intercept | Infinitely many solutions |
IB exam alert: If a question asks 'do these lines intersect?' first check gradients. Same gradient → parallel → no intersection.
Worked example — taxi cost
Company A charges 2 per km. Company B charges 3 per km. At what distance are the costs equal?
Step by step
- Set up equations. Cost A: C = 3 + 2d. Cost B: C = 1 + 3d.
- Set equal.
- Solve.
- Check cost.
Final answer
Costs are equal at 2 km ($7 each).