Concept 3.1
Two Lines, Three Stories
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
A pair of linear equations in two variables is represented by two lines. If the lines intersect at one point, that point is the unique solution; if they are parallel, there is no solution; if they coincide, there are infinitely many solutions.
The intuition
Each equation is a line: every point on it satisfies that equation. A solution of the pair must satisfy both, so it must lie on both lines.
That makes the question visual. Two lines can cross once, never, or everywhere. So a pair has one solution, none, or infinitely many, and nothing in between.
The attention
Look at where the two lines cross.
x + y = 5, x − y = 1 meet at (3, 2); they cut the x-axis at (5, 0) and (1, 0)
Look hereNot here
- Look here
- The point where the lines cross. Only the crossing point lies on both lines, so only it satisfies both equations.
- Not here
- Where one line meets an axis. Each intercept satisfies one equation only. (5, 0) is on x + y = 5, but 5 − 0 ≠ 1.
- What it teaches
- A solution to two conditions lives where both pictures overlap.