## 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*.

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 \(\left(0,-4\right).\)

\(4x-3y=12\) | |
---|---|

x | y |

\(3\) | \(0\) |

\(0\) | \(-4\) |