# Equation of a Line

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 $$\cfrac{1}{3}$$.

## Other Forms

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.

## Y-intercept / X-intercept

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 $$y = 12x - 24$$.