Q.Reduce the following equations into intercept form and find their intercepts on the axes.

Concept: Intercept form of a line is $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the $x$-intercept and $b$ is the $y$-intercept.

(i) $3x + 2y - 12 = 0$

Bring constant to RHS: $3x + 2y = 12$

Divide by 12: $\frac{3x}{12} + \frac{2y}{12} = 1 \implies \frac{x}{4} + \frac{y}{6} = 1$

Intercepts: $a = 4$, $b = 6$

(ii) $4x - 3y = 6$

Divide by 6: $\frac{4x}{6} - \frac{3y}{6} = 1 \implies \frac{x}{3/2} + \frac{y}{-2} = 1$

Intercepts: $a = \frac{3}{2}$, $b = -2$

(iii) $3y + 2 = 0$

Rewrite: $3y = -2 \implies y = -\frac{2}{3}$

This is a horizontal line; it never meets the $x$-axis. In intercept form: $\frac{x}{\infty} + \frac{y}{-2/3} = 1$ is not meaningful.

Intercepts: $x$-intercept does not exist, $y$-intercept is $-\frac{2}{3}$.

Concept First: Why Intercept Form?

The intercept form of a straight line is:

Here, $a$ is the $x$-intercept (where the line crosses the $x$-axis, so $y=0$), and $b$ is the $y$-intercept (where the line crosses the $y$-axis, so $x=0$).

Why is this useful? Because it tells you at a glance where the line cuts the axes — no need to solve two separate equations each time. Converting any linear equation into this form is just a matter of algebra: move the constant term to the right side, then divide through by that constant so the right side becomes $1$.

Step-by-Step Solutions

(i) $3x + 2y - 12 = 0$

1. Bring the constant term to the right side

2. Divide every term by 12 so the right side becomes $1$

3. Simplify each fraction

4. Read off the intercepts The $x$-intercept is $a = 4$, and the $y$-intercept is $b = 6$.

(ii) $4x - 3y = 6$

1. The equation is already in the form with constant on the right

2. Divide through by 6

3. Simplify

But this isn't quite the standard $\frac{x}{a} + \frac{y}{b} = 1$ — the minus sign means one intercept is negative. Rewrite as:

4. Read off the intercepts $x$-intercept $a = \frac{3}{2}$, $y$-intercept $b = -2$.

(iii) $3y + 2 = 0$

1. This equation has no $x$ term — it’s a horizontal line.

2. What does intercept form mean here?

   The line is parallel to the $x$-axis, so it never crosses it. There is no $x$-intercept. The $y$-intercept is simply $-\frac{2}{3}$.

3. Can we write it in $\frac{x}{a} + \frac{y}{b} = 1$?

   No — because $a$ would be infinite (the line never meets the $x$-axis). So we simply state the intercepts directly.