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.
x
y
0
0
Example
Find the intercepts of \(2x+y=6\)
We’ll fill in the figure below.
To find the x- intercept, let \(y=0\):
Substitute 0 for y.
Add.
Divide by 2.
The x-intercept is (3, 0).
To find the y- intercept, let \(x=0\):
Substitute 0 for x.
Multiply.
Add.
The y-intercept is (0, 6).
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 \(\left(0,-4\right).\)
