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Precalculus JumpStart

Anthony van Groningen

Section 5.5 Point-Slope Form

Suppose we want to find the equation of a line passing through a known point \(\mathcal{P}(x_0,y_0)\) with a known slope \(m\text{.}\) We need to determine an algebraic equation that we can use to determine if a different point \(\mathcal{Q}(x,y)\) is or is not on the line. If \(\mathcal{Q}\) is on the line, then we can calculate the (known) slope \(m\) from these two points using the slope formula (5.5)
\begin{equation*} m = \frac{y-y_0}{x-x_0}. \end{equation*}
Rewriting, we obtain the important point-slope form of the line
\begin{equation} y-y_0 = m(x-x_0).\tag{5.6} \end{equation}

Example 5.13. Using Point-Slope Form.

Let’s find an equation for the line passing through the points \((7,2)\) and \((-3,-9)\text{.}\)
  1. First, we need to determine its slope:
    \begin{equation*} m = \frac{2-(-9)}{-3-7} = \frac{11}{-10} = - \frac{11}{10}. \end{equation*}
  2. Next, we can use point-slope form (5.6) and either of the known points. Using \((7,2)\text{,}\)
    \begin{equation*} y - 2 = -\frac{11}{10}(x-7). \end{equation*}
It would not be unusual to rewrite the last equation by solving for \(y\text{.}\) In the case of the previous example, we would have
\begin{align*} y - 2 \amp= -\frac{11}{10}(x-7)\\ y \amp= -\frac{11}{10}(x-7) + 2\\ y \amp= -\frac{11}{10}x+\frac{77}{10} + 2\\ y \amp= -\frac{11}{10}x+\frac{77}{10} + \frac{20}{10}\\ y \amp= -\frac{11}{10}x+\frac{97}{10}\\ y \amp= -1.1 x+9.7 \end{align*}
In this final form we can easily see two things: the slope \(m = -1.1\) and the location of the \(y\)-intercept, for when \(x=0\text{,}\) \(y = 9.7\text{.}\) A linear equation in the form
\begin{equation} y = mx + b\tag{5.7} \end{equation}
is said to be in slope-intercept form. The slope of the line is \(m\) and the \(y\)-intercept is \((0,b)\text{.}\)

Example 5.14. Sketching a Line in Slope-Intercept Form.

Sketch a graph of the line with equation \(y = 3x + 1\text{.}\)
Solution.
The line has \(y\)-intercept at \((0,1)\) and slope \(m=3\text{.}\) Starting at the intercept, an increase of \(x\) by one unit will result in an increase of \(y\) by three units.
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