NCERT Solutions for Class 9 Maths Chapter 2 — Exercise Set 2.6#
Every equation of the form $y = a x + b$ has a straight-line graph. The number $a$ is the slope (steepness and direction of the line) and $b$ is the y-intercept (the point where the line crosses the y-axis).
1. Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’. #
(i) y = 4x, y = 2x, y = x
Pick a few $x$-values and find $y$:
| $x$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|
| $y = 4x$ | $-4$ | $0$ | $4$ | $8$ |
| $y = 2x$ | $-2$ | $0$ | $2$ | $4$ |
| $y = x$ | $-1$ | $0$ | $1$ | $2$ |
All three lines pass through the origin ($b = 0$). As $a$ increases from $1 \to 2 \to 4$, the line becomes steeper.
Role: $b = 0$ places the line through the origin; a larger $|a|$ makes the line steeper.
(ii) y = −6x, y = −3x, y = −x
| $x$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|
| $y = -6x$ | $6$ | $0$ | $-6$ | $-12$ |
| $y = -3x$ | $3$ | $0$ | $-3$ | $-6$ |
| $y = -x$ | $1$ | $0$ | $-1$ | $-2$ |
All three lines pass through the origin, and a negative $a$ makes the line slope downward from left to right.
Role: $a < 0$ reverses the slope direction (line goes down); a larger $|a|$ still means steeper.
(iii) y = 5x, y = −5x
| $x$ | $-1$ | $0$ | $1$ |
|---|---|---|---|
| $y = 5x$ | $-5$ | $0$ | $5$ |
| $y = -5x$ | $5$ | $0$ | $-5$ |
Both lines pass through the origin. The two lines are mirror images of each other in the x-axis (also in the y-axis).
Role: changing the sign of $a$ reflects the line across the axes; the magnitude of the slope is the same.
(iv) y = 3x − 1, y = 3x, y = 3x + 1
| $x$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|
| $y = 3x - 1$ | $-4$ | $-1$ | $2$ | $5$ |
| $y = 3x$ | $-3$ | $0$ | $3$ | $6$ |
| $y = 3x + 1$ | $-2$ | $1$ | $4$ | $7$ |
All three have the same slope $a = 3$, so they are parallel. They differ only in the y-intercept $b$, which shifts each line up or down.
Role: same $a$ ⇒ parallel lines; $b$ controls the vertical position (where the line crosses the y-axis).
(v) y = −2x − 3, y = −2x, y = 2x + 3
| $x$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|
| $y = -2x - 3$ | $-1$ | $-3$ | $-5$ | $-7$ |
| $y = -2x$ | $2$ | $0$ | $-2$ | $-4$ |
| $y = 2x + 3$ | $1$ | $3$ | $5$ | $7$ |
The first two lines have the same slope $a = -2$ and are therefore parallel. The third has slope $+2$, so it is not parallel to the other two — it is the mirror image of $y = -2x - 3$ in the x-axis.
Role: lines with the same $a$ are parallel regardless of $b$; changing the sign of $a$ tilts the line the other way.