## Identifying the Intercepts on a Graph

Every **linear equation** has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an **intercept of the line**.

### Definition: Intercepts of a Line

Each of the points at which a line crosses the \(x\text{-axis}\) and the \(y\text{-axis}\) is called an intercept of the line.

Let’s look at the graph of the lines shown in the figure below.

First, notice where each of these lines crosses the *x*– axis:

Figure: | The line crosses the x-axis at: | Ordered pair of this point |
---|---|---|

a. | 3 | (3,0) |

b. | 4 | (4,0) |

c. | 5 | (5,0) |

d. | 0 | (0,0) |

Do you see a pattern?

For each row, the *y-* coordinate of the point where the line crosses the *x-* axis is zero. The point where the line crosses the *x-* axis has the form \(\left(a,0\right)\); and is called the *x-intercept* of the line. The **x-** intercept occurs when y is zero.

Now, let’s look at the points where these lines cross the y-axis.

Figure: | The line crosses the y-axis at: | Ordered pair for this point |
---|---|---|

a. | 6 | (0,6) |

b. | -3 | (0,-3) |

c. | -5 | (0,-5) |

d. | 0 | (0,0) |

### Definition: *x-* intercept and *y-* intercept of a line

The \(x\text{-intercept}\) is the point, \(\left(a,0\right),\) where the graph crosses the \(x\text{-axis}.\) The \(x\text{-intercept}\) occurs when \(\text{y}\) is zero.

The \(y\text{-intercept}\) is the point, \(\left(0,b\right),\) where the graph crosses the \(y\text{-axis}.\)

The \(y\text{-intercept}\) occurs when \(\text{x}\) is zero.

## Example

Find the \(x\text{- and}\phantom{\rule{0.2em}{0ex}}y\text{-intercepts}\) of each line:

\(\phantom{\rule{0.2em}{0ex}}x+2y=4\) | |

\(\phantom{\rule{0.2em}{0ex}}3x-y=6\) | |

\(\phantom{\rule{0.2em}{0ex}}x+y=-5\) |

### Solution

The graph crosses the x-axis at the point (4, 0). | The x-intercept is (4, 0). |

The graph crosses the y-axis at the point (0, 2). | The x-intercept is (0, 2). |

The graph crosses the x-axis at the point (2, 0). | The x-intercept is (2, 0) |

The graph crosses the y-axis at the point (0, −6). | The y-intercept is (0, −6). |

The graph crosses the x-axis at the point (−5, 0). | The x-intercept is (−5, 0). |

The graph crosses the y-axis at the point (0, −5). | The y-intercept is (0, −5). |