Mathematics · Ch 4 — Straight Lines

Intercept - Form

4.3.5

Intercept - Form

The Intercept Form of a Line

When a line crosses the coordinate axes, the points where it meets the x-axis and y-axis are especially easy to work with. Suppose a line $L$ cuts the x-axis at a distance $a$ from the origin, and the y-axis at a distance $b$ from the origin. These distances are called the x-intercept and y-intercept, respectively.

Watch out

Intercepts are signed lengths. If the line meets the x-axis to the left of the origin, $a$ is negative. If it meets the y-axis below the origin, $b$ is negative. The intercept form works for all real values of $a$ and $b$ , except $a=0$ or $b=0$ (the line would then pass through the origin, and the form breaks down).

Since the line meets the x-axis at $(a, 0)$ and the y-axis at $(0, b)$ , we have two known points. The two-point form of a line gives us a direct way to write its equation.

Deriving the Intercept Form

Let the line pass through $P(a, 0)$ and $Q(0, b)$ . Using the two-point form:

\frac{y - 0}{x - a} = \frac{b - 0}{0 - a}

Simplify the right-hand side:

\frac{y}{x - a} = \frac{b}{-a}

Cross-multiply:

-ay = b(x - a)

Expand:

-ay = bx - ab

Bring all terms to one side:

bx + ay = ab

Now divide both sides by $ab$ (provided $a \neq 0$ and $b \neq 0$ ):

\frac{x}{a} + \frac{y}{b} = 1

\frac{x}{a} + \frac{y}{b} = 1

This is the intercept form of the equation of a line. The numbers $a$ and $b$ are the x-intercept and y-intercept, respectively.

Note

The intercept form is a special case of the general linear equation $Ax + By + C = 0$ . If you multiply through by $ab$ , you get $bx + ay - ab = 0$ , which is in general form with $A = b$ , $B = a$ , and $C = -ab$ .

Worked Example

Example 8: Find the equation of the line which makes intercepts $-3$ and $2$ on the x- and y-axes respectively.

Solution: Here $a = -3$ and $b = 2$ . Substitute directly into the intercept form:

\frac{x}{-3} + \frac{y}{2} = 1

Multiply through by the common denominator 6 to clear fractions:

-2x + 3y = 6

Or, rearranged:

-2x + 3y - 6 = 0

This is the required equation.

Tip

When an intercept is negative, the corresponding term in the intercept form becomes negative. For instance, $\frac{x}{-3}$ is the same as $-\frac{x}{3}$ . Always keep the sign with the intercept value.

Key Points to Remember

The intercept form $\frac{x}{a} + \frac{y}{b} = 1$ is valid only when both $a$ and $b$ are non-zero. If either intercept is zero, the line passes through the origin, and you must use another form (like slope-intercept or two-point).
The intercepts are the actual coordinates where the line meets the axes: $(a, 0)$ and $(0, b)$ . …

Figure 9.13Intercept form

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 9.13 is a simple coordinate-plane sketch that shows a straight line crossing both axes. The line is drawn with double arrows at its ends to indicate it extends infinitely in both directions. It meets the x-axis at the point $(a, 0)$ and the y-axis at the point $(0, b)$ . Two dashed segments are marked along the axes: one from the origin to $(a,0)$ along the x-axis, labelled with length $a$ ; another from the origin to $(0,b)$ along the y-axis, labelled with length $b$ . These are the intercepts — the distances (with sign) at which the line cuts the axes.

The physical idea is simple: if you know where a line hits the two axes, you can write its equation directly, without needing the slope. The intercepts $a$ and $b$ completely determine the line, provided neither is zero. The figure makes this geometric fact visual — the line is pinned down by its two axis-crossing points.

The textbook uses the two-point form of a line to derive the equation. Since the line passes through $(a,0)$ and $(0,b)$ , the slope is $\frac{b-0}{0-a} = -\frac{b}{a}$ . Using point-slope form with $(a,0)$ :

y - 0 = -\frac{b}{a}(x - a)

Multiplying out: $y = -\frac{b}{a}x + b$ . Rearranging gives $\frac{x}{a} + \frac{y}{b} = 1$ .

\frac{x}{a} + \frac{y}{b} = 1

This is the intercept form of the equation of a line. Here $a$ is the x-intercept (the x-coordinate where the line meets the x-axis) and $b$ is the y-intercept (the y-coordinate where the line meets the y-axis). Both $a$ and $b$ can be positive, negative, or zero — but if either is zero, the line passes through the origin and the intercept form fails (the corresponding term becomes undefined). In that case you use a different form. …

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