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:

\( y = 4x - 1\)

Equation of a line
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.

X and Y Intercept of a line

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 \( y = 12x - 24 \).