Equations from Intercepts

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Idea

The intercepts are the points where a line crosses the axes. They are ordinary points, but they are the easiest ones to read from a graph, because each one has a zero in it.

A line crossing the $x$ -axis at $a$ passes through $(a, 0)$ . Crossing the $y$ -axis at $b$ means it passes through $(0, b)$ . Two points give a line, so the intercepts alone are enough to find the equation.

Rule

A line with $x$ -intercept $a$ and $y$ -intercept $b$ (neither zero) has the intercept form

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

Each intercept sits in the denominator under its own variable. Multiplying through by $ab$ clears it to standard form.

How it is used

There are three routine tasks.

Given	Do this
Both intercepts	drop them straight into $\dfrac{x}{a} + \dfrac{y}{b} = 1$
An equation, want the intercepts	set $y = 0$ for $a$ ; set $x = 0$ for $b$
Both intercepts, want the slope	$m = -\dfrac{b}{a}$ , falling whenever $a$ and $b$ share a sign

The intercept form does not always work.

If the line passes through the origin, both intercepts are $0$ , and you cannot divide by $0$ . If the line is horizontal or vertical, one of the intercepts does not exist. The previous two topics cover those lines.

To read intercepts from a graph, look only at where the line crosses the axes. To draw a line from its equation, find both intercepts, mark them, and join them.

Worked example

Find the equation of the line whose $x$ -intercept is $4$ and whose $y$ -intercept is $-2$ . The intercepts give the points $(4, 0)$ and $(0, -2)$ . Put them into the intercept form, with each intercept under its own variable:

\frac{x}{4} + \frac{y}{-2} = 1

Multiply every term by $4$ to clear the denominators, giving $x - 2y = 4$ . Collect on one side for standard form:

x - 2y - 4 = 0

Explore

Put A on the $x$ -axis and B on the $y$ -axis. The green circles land on top of them, because those points are the intercepts. Slide one intercept along its axis and watch the equation change. Then move an intercept close to the origin and see what happens.

A line through two points

Try this. Move the two points and compare the equation. Equal x values produce a vertical line; equal points cannot define one.

Point A · x-2Point A · y3Point B · x3Point B · y1

y = -0.4x + 2.2 · Slope = -0.4

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

A line crosses at

(4,0)

and

(0,2)

. Which equation fits?

$x/2+y/4=1$

$x/4+y/2=0$

$x/4+y/2=1$

Hint 1 · Find a starting point

An intercept lies on an axis, so one coordinate is zero.

Hint 2 · Take the next step

Test the candidate at x = 4, y = 0, then at x = 0, y = 2.

Show the reasoning

Answer: $x/4+y/2=1$

Both intercepts satisfy $x/4+y/2=1$ . The denominators identify the axis crossings; their order matters.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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