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Find the X and Y Intercepts of a Line Using Algebra
> ... Math > Slopes, Intercepts, Equation of a Straight Line > Find the X and Y Intercepts of a Line Using Algebra

Introduction

In this lesson, you will learn how to find the x- and y-intercepts using algebra.

This video illustrates the lesson material below. Watching the video is optional.

Find the X- and Y-Intercepts Using Algebra

When the equation is written in the slope-intercept form (\(y=mx+b\)), you can find the y-intercept by looking at the equation. The value of b is the y-intercept. This is because the y-intercept is when the x value equals 0.

\begin{align*} y&=mx+b &\color{red}\small\text{Slope-intercept form}\\\\ y&=m(0)+b &\color{red}\small\text{Substitute x=0}\\\\ y&=0+b &\color{red}\small\text{Multiply the slope & 0}\\\\ y&=b &\color{red}\small\text{Simplify so y = b}\end{align*}

So, when \(x = 0\), the y-intercept is b.

To find the x-intercept, set \(y = 0\) and solve the equation for x. This is because when \(y = 0\) the line crosses the x-axis.

Example 1
Use the following equation to find the x-intercept and the y-intercept: \(y=\frac{3}{4}x-2\).

  • The x-intercept is found whenever \(y=0\)
  • The y-intercept is found whenever \(x=0\)

Start by finding the x-intercept:
\begin{align*}
y&=\frac{3}{4}x-2 &\color{red}\small\text{Given equation}\\\\
0&=\frac{3}{4}x-2 &\color{red}\small\text{Set y = 0 to find the x-intercept}\\\\
0 \color{red}\mathbf{+2} &=\frac{3}{4}x-2 \color{red}\mathbf{+2} &\color{red}\small\text{Additive inverse}\\\\
2 &=\frac{3}{4}x &\color{red}\small\text{Simply both sides}\\\\
2 \color{red}\mathbf{ (\frac{4}{3})} &=\frac{3}{4}x \color{red}\mathbf{(\frac{4}{3})} &\color{red}\small\text{Multiply both sides by the reciprocal}\\\\
\frac{8}{3} &= x &\color{red}\small\text{Simplify both sides}
\end{align*}

The x-intercept is \((\frac{8}{3},0)\), or 2.67.

Now, calculate the y-intercept:

\begin{align*}
y&=\frac{3}{4}x-2 &\color{red}\small\text{Given equation}\\\\
y&=\frac{3}{4}(0)-2 &\color{red}\small\text{Set x = 0 to find the y-intercept}\\\\
y&=0-2 &\color{red}\small\text{Multiply \(\frac{3}{4} (0)\) = 0}\\\\
y&=-2 &\color{red}\small\text{Simplify by subtracting}
\end{align*}

In the equation \(y=mx+b\), the b in the equation is the y-intercept.

The y-intercept is -2 which can be written as an ordered pair \((0, -2)\).

Example 2
Use the following equation to find the x-intercept and the y-intercept: \(3y-6x=12\).

In Example 1, the equation was in slope-intercept form. This equation is in a different format. However, you can still use the same principles to find the x- and y-intercepts. Remember:

  • The x-intercept is found whenever \(y=0\)
  • The y-intercept is found whenever \(x=0\)

Start by finding the x-intercept:

\begin{align*}
3y - 6x&=12 &\color{red}\small\text{Given equation}\\\\
3(0) - 6x&=12 &\color{red}\small\text{Set y = 0 to find the x-intercept}\\\\
-6x & =12 &\color{red}\small\text{Multiply 3(0) = 0}\\\\
\frac{-6x}{\color{red}\mathbf{-6}} &=\frac{12}{\color{red}\mathbf{-6}} &\color{red}\small\text{Divide both sides by -6}\\\\
x &=-2 &\color{red}\small\text{Simplify both sides}
\end{align*}

The x-intercept is \((-2,0)\).

Now, calculate the y-intercept:

\begin{align*}
3y - 6x&=12 &\color{red}\small\text{Given equation}\\\\
3y - 6(0)&=12 &\color{red}\small\text{Set x = 0 to find the y-intercept}\\\\
3y & =12 &\color{red}\small\text{Multiply -6(0) = 0}\\\\
\frac{3y}{\color{red}\mathbf{3}} &=\frac{12}{\color{red}\mathbf{3}} &\color{red}\small\text{Divide both sides by 3}\\\\
y &=4 &\color{red}\small\text{Simplify both sides}
\end{align*}

Things to Remember


  • To find x-intercept, set \(y = 0\) and solve for x. The point will be \((x, 0)\).
  • To find y-intercept, set \(x = 0\) and solve for y. The point will be \((0, y)\).

Practice Problems

1. Find the y-intercept of the line: (
Solution
x
Solution: \(y=-9\)
)
\({\text{y}}=-3{\text{x}}-9\)2. Find the x-intercept of the line: (
Solution
x
Solution: \(x=3\)
)
\({\text{y}}=-4{\text{x}}+12\)3. Find the y-intercept of the line: (
Solution
x
Solution: \(y=9\)

Details:
To find the y-intercept, set \({\color{Red}x = 0}\) and solve for y. The point will be \((0,y)\):

\(y − 9 = 3{\color{Red}x}\)

Substitute \({\color{Red}0}\) in for \({\color{Red}x}\):

\(y − 9 = 3 {\color{Red}(0)}\)

Multiply 3 times 0, which gives you:

\(y − 9 = {\color{Red}0}\)

Then add 9 to both sides to isolate y:

\(y − 9 {\color{Red}+9} = 0 {\color{Red}+9}\)

Which gives you:

\(y = 9\)

So the y-intercept is \((0,9)\).
)
\({\text{y}} − 9 = 3x\)4. Find the x-intercept of the line: (
Solution
x
Solution: \(x=6\)
)
\({\text{y}} + 12 = 2x\)5. Find the y-intercept of the line: (
Solution
x
Solution: \(y=-4\)
)
\({\text{x}}+6{\text{y}}=-24\)6. Find the x-intercept of the line: (
Solution
x
Solution: \(x=-4\)

Details:
To find the x-intercept:, set \({\color{Red}y = 0}\) and solve for x. The point will be \((x,0)\):

\(5{\text{x}}+4{\text{y}}=-20\)

Substitute \({\color{Red}0}\) in for y:

\(5{\text{x}}+4{\color{Red}(0)}=-20\)

Multiply 4 times 0 which gives you:

\(5{\text{x}}+{\color{Red}0}=-20\)

Add 5x to 0:

\(\frac{1}{5}(5{\text{x}})\) = \(5{\text{x}}=-20\)

Then multiply both sides by \(\dfrac{1}{5}\) (or divide both sides by 5, both will give you the same solution):

\(-20\dfrac{1}{5}\)

Which gives you:

\({\text{x}}=-4\)

So the x-intercept is \((-4,0)\).
)
\(5{\text{x}}+4{\text{y}}=-20\)

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