Q.Find the equation of the line, which makes intercepts $-3$ and $2$ on the x- and y-axes respectively.

Concept: Intercept Form of a Line — When a line cuts the x-axis at $(a,0)$ and the y-axis at $(0,b)$, its equation is $\frac{x}{a} + \frac{y}{b} = 1$.

Here, $a = -3$ and $b = 2$.

Substitute directly into the intercept form:

$\frac{x}{-3} + \frac{y}{2} = 1$ …

The key idea here is the intercept form of a straight line. When a line cuts the x-axis at a point $(a, 0)$ and the y-axis at $(0, b)$, the numbers $a$ and $b$ are called the x-intercept and y-intercept respectively. The beauty of this form is that it directly uses these two pieces of information to write the equation — no slope calculation needed.

Why does this work? If a line passes through $(a, 0)$ and $(0, b)$, its slope is $\frac{b - 0}{0 - a} = -\frac{b}{a}$. Using the point-slope form with point $(0, b)$ gives $y - b = -\frac{b}{a}(x - 0)$, which simplifies to $\frac{x}{a} + \frac{y}{b} = 1$. So the intercept form is just a tidy rearrangement of the standard two-point form.

Now, let’s apply it step by step.

1. Identify the intercepts. The problem states: intercepts $-3$ on the x-axis and $2$ on the y-axis. So $a = -3$ and $b = 2$. Note that the negative sign on the x-intercept means the line crosses the x-axis to the left of the origin.

2. Plug into the intercept form.

3. Simplify the equation. Multiply both sides by the least common multiple of the denominators, which is $6$, to clear fractions:

$-2x + 3y = 6$ …