How to Find the Intercepts of a Line

Problem

2x + 3y = 12

Answer

x\text{-intercept } (6, 0),\quad y\text{-intercept } (0, 4)

An intercept is where the line crosses an axis. On the x-axis y is zero, and on the y-axis x is zero. Each substitution turns the equation into one step.

Step-by-step solution

Find the x-intercept by setting y = 0.
$2x + 3(0) = 12 \Rightarrow 2x = 12$
Why: The graph crosses the x-axis where y is zero.
Solve for x.
$x = 6$
Why: Divide both sides by 2.
Find the y-intercept by setting x = 0.
$2(0) + 3y = 12 \Rightarrow 3y = 12 \Rightarrow y = 4$
Why: The graph crosses the y-axis where x is zero.
Write each intercept as a point.
$(6, 0)\ \text{and}\ (0, 4)$
Why: An intercept is a point, so it needs both coordinates.

Common mistakes

Writing 6 as the x-intercept instead of the point (6, 0).
Swapping the coordinates, so the x-intercept becomes (0, 6).
Setting both variables to zero at once, which gives 0 = 12 and no point.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$x + y = 5$

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$(5, 0),\ (0, 5)$
$3x + 2y = 6$

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$(2, 0),\ (0, 3)$
$2x - y = 8$

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$(4, 0),\ (0, -8)$
$4x + y = 8$

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$(2, 0),\ (0, 8)$
$x - 2y = 6$

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$(6, 0),\ (0, -3)$