A linear relation can be described using the equation of a line in slope-intercept form:

\( y = m(x) + b\)

where \(m\) is the slope and \(b\) is the y-intercept.

The **slope** represents the steepness of the line. The **y-intercept** is the point where the line crosses the __y-axis__.

In the diagram below, the line crosses the y-axis at \(y=-1\) (y-int). Everytime x increases by 1,
y increases by 4 (__slope__). The equation of the line is:

\( y = 4x - 1\)

Write the equation for a line with a slope of \(-2\) and a y-int of \( \cfrac{1}{3} \).

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There are a few other ways we can write an equation of a line. One is the **point-slope form** and another is **standard form**.

The equation for point-slope form is:

\( y - y_0 = m (x-x_0)\)

where \(m\) is the slope and \(x_0, y_0\) is a point on the line.

The equation for standard form is:

\( Ax + By = C\)

where \(A, B, C\) are coefficients wiht \(A, B \ne 0\). If we re-arrange the equation for \(y\) we get:

\( y = \cfrac{-A}{B}x + \cfrac{C}{B}\)

From here we can see the slope is:

\( m = \cfrac{-A}{B}\)

Tyically, \(A, B, C\) are integers, \(A\) is positive and \(A, B, C\) do not share common factors.

Let's dig a little deeper into the **y-intercept** and also the **x-intercept**.

Intercepts are points on a line where the line __crosses__ the axes. The __y-intercept__ is where the line crosses the __y-axes__. The __x-intercept__ is where the line crosses the __x-axes__. See the figure below.

Notice that the __x-value__ of the __y-int__ and the __y-value__ of the __x-int__ are 0. So, to solve for either the x-intercept, or the y-intercept; we just have to set the __opposite__ value to zero.

Let's fine the intercepts for the equation \(y = (-3)x - 4\).

To find the x-intercept, set \(y = 0\). | To find the y-intercept, set \(x = 0\). |
---|---|

\( 0 = -3(x) - 4\) | \( y = -3(0) - 4\) |

To find the x-intercept, solve for \(x\). | To find the y-intercept, solve for \(y\). |

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

\(x = \cfrac{-4}{3}\) | |

x-intercept | y-intercept |

\( (\cfrac{-4}{3},0) \) | \( ( 0,-4) \) |

Determine the intercepts of the line \( y = 12x - 24 \).

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