Two Variables, One Line
Solving by graph
Solve a pair of equations by drawing both lines.
A solution of the pair is a point on both lines. Draw both lines and look where they meet.
Three pictures, three answers:
- lines cross at one point: one solution. x + 3y = 6 and 2x – 3y = 12 cross at (6, 0);
- lines are parallel: no solution. x + 2y – 4 = 0 and 2x + 4y – 12 = 0 both have slope –1/2;
- lines lie on top of each other: infinitely many. 2x + 3y = 9 and 4x + 6y = 18 are the same line.
Equal slopes mean parallel. y = mx + d1 and y = mx + d2 can only meet if d1 = d2, and then they are the same line.
Comparing two phone plans. Plan A costs 50 + 0.2t rupees and Plan B costs 30 + 0.3t for t minutes. They cost the same at t = 200 minutes. Below 200, Plan B is cheaper; above it, Plan A. The crossing point is where the better choice changes.
A graph shows the answer; algebra pins it down exactly. Always check the point in both equations.
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