Point-Slope Form

Want some other practice with lines?

• Introduction to the Slope of a Line
• Practice with Slope
• Graphing Lines
• Finding Equations of Lines

Suppose a line has slope $\,m\,$ and passes through a known point $\,(x_1,y_1)\,.$ That is, we know the slope of the line and we know a point on the line.

We can get an equation that is ideally suited to these two pieces of information. This equation is appropriately called the point-slope form of a line.

Here's what to do:

• Recall that $\,(x_1,y_1)\,$ is a known point on a line with slope $\,m\,.$
• Let $\,(x,y)\,$ denote any other point on the line.
• Now, we have two points:  the known point $\,(x_1,y_1)\,$ and a ‘generic’ point $\,(x,y)\,.$
• The slope of the line, computed using these two points, must equal $\,m\,.$
• Using the slope formula, we have: $$\cssId{s15}{m = \frac{y-y_1}{x-x_1}}$$ or, equivalently, $$\cssId{s17}{y - y_1 = m(x - x_1)}$$

This gives us an extremely useful equation of a line, as summarized below:

Important Things to Know About Point-Slope Form:

• The variables in the equation $\,y - y_1 = m(x - x_1)\,$ are $\,x\,$ and $\,y\,.$ That is, this is an equation in two variables, $\,x\,$ and $\,y\,.$ Thus, its solution set is the set of all ordered pairs $\,(x,y)\,$ that make it true.
• For a given equation:
  • the number $\,m\,$ is a constant (a specific number) that represents the slope of the line;
  • the number $\,x_1\,$ (read as ‘ex sub one’) is a constant that represents the $\,x$-value of the known point;
  • the number $\,y_1\,$ (read as ‘wye sub one’) is a constant that represents the $\,y$-value of the known point
• As we vary the values of $\,m\,,$ $\,x_1\,,$ and $\,y_1\,,$ we get lots of different equations. Here are some of them: $$\begin{gather} \cssId{s38}{y - 2 = 5(x - 3)}\cr \cssId{s39}{(m = 5\,,\ x_1 = 3\,, \ \text{and}\ \ y_1 = 2)}\cr\cr \cssId{s40}{y - \frac12 = \sqrt{2}(x - 3.4)}\cr \cssId{s41}{(m = \sqrt2\,,\ x_1 = 3.4\,, \ \ \text{and}\ \ y_1 = \frac12)}\cr\cr \cssId{s42}{y = 5(x + 1)}\cr \cssId{s43}{\text{Rewrite the equation as:}}\cr \cssId{s44}{y - 0 = 5(x - (-1))}\cr \cssId{s45}{\text{Thus, we see that:}}\cr \cssId{s46}{m = 5\,, \ x_1 = -1\,, \ \text{and}\ y_1 = 0} \end{gather} $$
• So, even though the equation $$y - y_1 = m(x - x_1)$$ uses five different ‘letters’ $$\cssId{s48}{y\,,\ y_1\,,\ m\,,\ x\,,\ \text{and}\ \, x_1}$$ they play very different roles:
  • $\,x\,$ and $\,y\,$ are the variables; they determine the nature of the solution set
  • $\,m\,,$ $\,x_1\,$ and $\,y_1\,$ are called parameters; they are constant in any particular equation, but vary from equation to equation.
• This is another beautiful example of the power/compactness of the mathematical language! The single equation $$y - y_1 = m(x - x_1)$$ actually describes an entire family of equations, which has infinitely-many members. We get the members of this family by choosing real numbers $\,m\,,$ $\,x_1\,$ and $\,y_1\,$ to plug in.
• If you know the slope of a line and the $\,y$-intercept, then it's probably easiest to use slope-intercept form. But, if you know the slope of a line and a point that isn't the $\,y$-intercept, then it's easiest to use point-slope form.
• Remember—just as expressions have lots of different names, so do sentences. Every non-vertical line can be written in any of these forms:
  • point-slope form: $$\cssId{s66}{y - y_1 = m(x - x_1)}$$
  • slope-intercept form: $$\cssId{s68}{y = mx + b}$$
  • general form: $$\cssId{s70}{ax + by + c = 0}$$
• Here's an example. (Make sure you convince yourself that these are equivalent equations!)
  • point-slope form: $$y - 2 = 5(x - 3)$$
  • slope-intercept form: $$y = 5x -13$$
  • general form: $$5x - y - 13 = 0$$

Example