How do we find solutions from the graph? What can we say about the number of solutions when the two lines (i) intersect at a point, (ii) are parallel, (iii) are coincident?
Step-by-step solution
Idea: Each line is the set of solutions of its own equation. A solution of the pair must satisfy both equations, so it must be on both lines: look for common points.
- Solutions of the pair are the points common to both lines, so read off where the lines meet.½ mark
- (i) Two lines that intersect share exactly one point: a unique solution, e.g. 2x + y = 6 and x − y = 2 meeting at (83, 23).½ mark
- (ii) Parallel lines have no common point: no solution, e.g. x − 2y = 4 and 2x − 4y = 6.½ mark
- (iii) Coincident lines share every point: infinitely many solutions, e.g. x + y = 4 and 2x + 2y = 8.½ mark
Check: These match the ratio rules: a1a2 ≠ b1b2 (intersect), a1a2 = b1b2 ≠ c1c2 (parallel), all three ratios equal (coincident) ✓.
Answer to write in the exam
Solution of the pair = point common to both lines
(i) Intersecting lines ⇒ exactly one solution
(ii) Parallel lines ⇒ no solution
(iii) Coincident lines ⇒ infinitely many solutions
Common mistakes that cost marks
- Saying parallel lines have infinitely many solutions because they “go on forever”.
- Giving a rough reading from a sketch as exact; always check the point in both equations.
How this can come in the exam
If the graphs of two linear equations are coincident lines, the pair has
- no solution
- exactly one solution
- exactly two solutions
- infinitely many solutions
Show answer
(D) infinitely many solutions
Every point of the line is common to both.
Try one yourself
Two lines have slopes 2 and −1. How many solutions does their pair of equations have?
Show answer
Different slopes ⇒ the lines intersect ⇒ exactly one solution.
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