Table of Contents
In the study of linear functions, one very convenient way to write the equation of a line is intercept form. This form makes it easy to read where the line crosses the coordinate axes.
What “intercept form” means
For a non-vertical, non-horizontal line that crosses both axes, the intercept form is
$$
\frac{x}{a} + \frac{y}{b} = 1,
$$
where:
- $a$ is the $x$-intercept, the point where the line crosses the $x$-axis: $(a, 0)$.
- $b$ is the $y$-intercept, the point where the line crosses the $y$-axis: $(0, b)$.
By definition of intercepts:
- At the $x$-intercept, $y = 0$.
- At the $y$-intercept, $x = 0$.
In this form, you can immediately see those intercepts:
- Plug in $y = 0$:
$$
\frac{x}{a} + \frac{0}{b} = 1 \quad\Rightarrow\quad \frac{x}{a} = 1 \quad\Rightarrow\quad x = a.
$$ - Plug in $x = 0$:
$$
\frac{0}{a} + \frac{y}{b} = 1 \quad\Rightarrow\quad \frac{y}{b} = 1 \quad\Rightarrow\quad y = b.
$$
So the intercept form is built exactly so that the constants $a$ and $b$ are the intercepts.
When intercept form is useful
Intercept form is especially useful when:
- You know (or want to emphasize) both intercepts.
- You want a quick sketch of a line: plot $(a, 0)$ and $(0, b)$, then draw the line through them.
- You want to see clearly how the intercepts change when you change the equation.
It is not always the most convenient form for algebraic manipulation, but it is very geometric.
Converting between slope–intercept form and intercept form
You will often encounter lines in slope–intercept form:
$$
y = mx + c,
$$
where $m$ is the slope and $c$ is the $y$-intercept.
From slope–intercept form to intercept form
Suppose a line is given by $y = mx + c$ and has both intercepts (so $m \neq 0$ and $c \neq 0$).
- Find the $y$-intercept: This is already $c$, so the $y$-intercept is $(0, c)$. Thus $b = c$.
- Find the $x$-intercept: Set $y = 0$ and solve for $x$:
$$
0 = mx + c \quad\Rightarrow\quad mx = -c \quad\Rightarrow\quad x = -\frac{c}{m}.
$$
So the $x$-intercept is $\left(-\frac{c}{m}, 0\right)$, and $a = -\frac{c}{m}$.
- Write the intercept form:
$$
\frac{x}{a} + \frac{y}{b} = 1
\quad\Rightarrow\quad
\frac{x}{-\frac{c}{m}} + \frac{y}{c} = 1.
$$
You can simplify if you like, but the key is that $a$ and $b$ are the intercepts you just found.
Example
Rewrite $y = 2x + 6$ in intercept form.
- $y$-intercept: $c = 6$, so $b = 6$.
- $x$-intercept: set $y = 0$:
$$
0 = 2x + 6 \quad\Rightarrow\quad 2x = -6 \quad\Rightarrow\quad x = -3.
$$
So $a = -3$.
Intercept form:
$$
\frac{x}{-3} + \frac{y}{6} = 1.
$$
You can also write it as
$$
-\frac{x}{3} + \frac{y}{6} = 1,
$$
which is equivalent.
From standard or general form to intercept form
A common linear equation form is
$$
Ax + By = C
$$
with $A$ and $B$ not both zero.
- Divide every term by $C$ (assuming $C \neq 0$):
$$
\frac{Ax}{C} + \frac{By}{C} = 1.
$$
- Rewrite the fractions as
$$
\frac{x}{\frac{C}{A}} + \frac{y}{\frac{C}{B}} = 1
$$
(when $A \neq 0$ and $B \neq 0$).
So:
- $a = \dfrac{C}{A}$ is the $x$-intercept.
- $b = \dfrac{C}{B}$ is the $y$-intercept.
Example
Rewrite $3x + 4y = 12$ in intercept form.
- Divide by $12$:
$$
\frac{3x}{12} + \frac{4y}{12} = 1.
$$
- Simplify:
$$
\frac{x}{4} + \frac{y}{3} = 1.
$$
So $a = 4$ and $b = 3$, meaning the intercepts are $(4, 0)$ and $(0, 3)$.
From intercept form to slope–intercept form
Start with
$$
\frac{x}{a} + \frac{y}{b} = 1,
$$
with $a \neq 0$ and $b \neq 0$.
- Isolate $\frac{y}{b}$:
$$
\frac{y}{b} = 1 - \frac{x}{a}.
$$
- Multiply both sides by $b$:
$$
y = b\left(1 - \frac{x}{a}\right) = b - \frac{b}{a}x.
$$
- Rearrange slightly to match $y = mx + c$:
$$
y = \left(-\frac{b}{a}\right)x + b.
$$
So:
- Slope: $m = -\dfrac{b}{a}$.
- $y$-intercept: $c = b$.
Example
Convert $\dfrac{x}{5} + \dfrac{y}{-2} = 1$ to slope–intercept form.
Here $a = 5$ and $b = -2$.
Using the formula:
- $m = -\dfrac{b}{a} = -\dfrac{-2}{5} = \dfrac{2}{5}$.
- $c = b = -2$.
So
$$
y = \frac{2}{5}x - 2.
$$
You can also derive this directly:
$$
\frac{x}{5} + \frac{y}{-2} = 1
\quad\Rightarrow\quad
\frac{x}{5} - \frac{y}{2} = 1
\quad\Rightarrow\quad
-\frac{y}{2} = 1 - \frac{x}{5}
\quad\Rightarrow\quad
y = -2\left(1 - \frac{x}{5}\right) = \frac{2}{5}x - 2.
$$
Graphing a line from intercept form
To graph a line given in intercept form, you do not need to calculate the slope first.
Given
$$
\frac{x}{a} + \frac{y}{b} = 1,
$$
- Read off the intercepts:
- $x$-intercept: $(a, 0)$
- $y$-intercept: $(0, b)$
- Plot these two points.
- Draw the straight line through them, extending in both directions.
Example
Graph $\dfrac{x}{-4} + \dfrac{y}{2} = 1$.
- $x$-intercept: $a = -4 \Rightarrow (-4, 0)$
- $y$-intercept: $b = 2 \Rightarrow (0, 2)$
Plot $(-4, 0)$ and $(0, 2)$, then draw the line.
Slope in terms of intercepts
From the conversion above, for
$$
\frac{x}{a} + \frac{y}{b} = 1,
$$
the slope is
$$
m = -\frac{b}{a}.
$$
This matches the idea of slope as “rise over run” between the intercept points $(a, 0)$ and $(0, b)$:
$$
m = \frac{\text{change in } y}{\text{change in } x}
= \frac{b - 0}{0 - a}
= \frac{b}{-a}
= -\frac{b}{a}.
$$
So if you know both intercepts, you can find the slope immediately from this formula.
Special cases and limitations
Intercept form $\dfrac{x}{a} + \dfrac{y}{b} = 1$ assumes the line crosses both axes.
- If $a = 0$, the line would “intersect” the $x$-axis at $x = 0$, but then the equation no longer has the form $\dfrac{x}{a} + \dfrac{y}{b} = 1$ with finite $a$.
- If $b = 0$, similarly there is no meaningful $\dfrac{y}{b}$ term.
Important special lines:
- Vertical lines: $x = k$. These have no $y$-intercept if $k \neq 0$, so they cannot be written in intercept form.
- Horizontal lines: $y = k$. These have no $x$-intercept if $k \neq 0$, so they also do not fit the intercept form with both $a$ and $b$ finite.
In those cases, it is better to use other forms (like $x = k$ or $y = k$ directly, or slope–intercept form when possible).
Finding an equation in intercept form from two intercepts
Often a problem will give you the intercepts and ask for the equation.
If you are told:
- The line has $x$-intercept $(a, 0)$.
- The line has $y$-intercept $(0, b)$.
Then you can directly write:
$$
\frac{x}{a} + \frac{y}{b} = 1.
$$
Example
Find the equation of the line with $x$-intercept $(-2, 0)$ and $y$-intercept $(0, 5)$ in intercept form.
Here:
- $a = -2$
- $b = 5$
So the equation is
$$
\frac{x}{-2} + \frac{y}{5} = 1.
$$
You can leave it like this, or rearrange to another linear form if needed.
Summary of key points
- Intercept form of a line:
$$
\frac{x}{a} + \frac{y}{b} = 1
$$
where $a$ and $b$ are the $x$- and $y$-intercepts. - Intercepts:
- $x$-intercept: $(a, 0)$
- $y$-intercept: $(0, b)$
- Slope in terms of intercepts:
$$
m = -\frac{b}{a}.
$$ - Easy graphing: plot the two intercepts and draw the line.
- Only lines that cross both axes (not vertical or horizontal lines missing one intercept) can be written in this intercept form.