Mathematics » Introducing Graphs » Graphing with Intercepts

Finding the Intercepts from an Equation of a Line

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Finding the Intercepts from an Equation of a Line

Recognizing that the \(x\text{-intercept}\) occurs when \(y\) is zero and that the \(y\text{-intercept}\) occurs when \(x\) is zero gives us a method to find the intercepts of a line from its equation. To find the \(x\text{-intercept,}\) let \(y=0\) and solve for \(x.\) To find the \(y\text{-intercept},\) let \(x=0\) and solve for \(y.\)

Definition: Find the x and y from the Equation of a Line

Use the equation to find:

  • the x-intercept of the line, let \(y=0\) and solve for x.
  • the y-intercept of the line, let \(x=0\) and solve for y.
xy
 0
0 

Example

Find the intercepts of \(2x+y=6\)

We’ll fill in the figure below.

Finding the Intercepts from an Equation of a Line

To find the x- intercept, let \(y=0\):

 Finding the Intercepts from an Equation of a Line
Substitute 0 for y.Finding the Intercepts from an Equation of a Line
Add.Finding the Intercepts from an Equation of a Line
Divide by 2.Finding the Intercepts from an Equation of a Line
The x-intercept is (3, 0). 

To find the y- intercept, let \(x=0\):

 Finding the Intercepts from an Equation of a Line
Substitute 0 for x.Finding the Intercepts from an Equation of a Line
Multiply.Finding the Intercepts from an Equation of a Line
Add.Finding the Intercepts from an Equation of a Line
The y-intercept is (0, 6). 

Finding the Intercepts from an Equation of a Line

The intercepts are the points \(\left(3,0\right)\) and \(\left(0,6\right)\).

Example

Find the intercepts of \(4x-3y=12.\)

Solution

To find the \(x\text{-intercept,}\) let \(y=0.\)

 \(4x-3y=12\)
Substitute 0 for \(y.\)\(4x-3·0=12\)
Multiply.\(4x-0=12\)
Subtract.\(4x=12\)
Divide by 4.\(x=3\)

The \(x\text{-intercept}\) is \(\left(3,0\right).\)

To find the \(y\text{-intercept},\) let \(x=0.\)

 \(4x-3y=12\)
Substitute 0 for \(x.\)\(4·0-3y=12\)
Multiply.\(0-3y=12\)
Simplify.\(-3y=12\)
Divide by −3.\(y=-4\)

The \(y\text{-intercept}\) is \(\left(0,-4\right).\)

The intercepts are the points \(\left(-3,0\right)\) and \(\left(0,-4\right).\)

\(4x-3y=12\)
xy
\(3\)\(0\)
\(0\)\(-4\)

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