Point-Slope Form

Suppose a line has slope $\,m\,$ and passes through the 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, as follows:

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:

$\displaystyle\,m = \frac{y-y_1}{x-x_1}\,$

or, equivalently,

$\,y - y_1 = m(x - x_1)\,$

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

POINT-SLOPE FORM line with slope $\,m\,$, passing through $\,(x_1,y_1)\,$

The graph of the equation

$$y - y_1 = m(x - x_1)$$ is a line with slope $\,m\,$ that passes through the point $\,(x_1,y_1)\,$.
Since this equation is ideally suited to the situation where you know a point and a slope,
it is appropriately called point-slope form.

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:

$y - 2 = 5(x - 3)$	($\,m = 5\,$, $\,x_1 = 3\,$, and $\,y_1 = 2\,$)
$y - \frac12 = \sqrt{2}(x - 3.4)$	($\,m = \sqrt2\,$, $\,x_1 = 3.4\,$, and $\,y_1 = \frac12\,$)
$y = 5(x + 1)$	Rewrite the equation as: $\,y - 0 = 5(x - (-1))\,$ Thus, we see that: $\,m = 5\,$, $\,x_1 = -1\,$, and $\,y_1 = 0\,$

So, even though the equation [beautiful math coming... please be patient] $\,y - y_1 = m(x - x_1)\,$
uses five different ‘letters’ ($\,y\,$, $\,y_1\,$, $\,m\,$, $\,x\,$, 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: $y - y_1 = m(x - x_1)$
- slope-intercept form: $y = mx + b$
- general form: $ax + by + c = 0$
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:

Question:
Write the point-slope equation of the line with slope $\,5\,$ that passes through the point $\,(3,-2)\,$.
Then, write the line in $\,y = mx + b\,$ form.

Solution:
Here,

$\,(x_1,y_1)\,$ is $\,(3,-2)\,$ and $\,m = 5\,$.
Substitution into $\,y - y_1 = m(x-x_1)\,$ gives:

$$ \underset{y}{\strut \overset{y}{\overbrace{\strut\ \ \ y\ \ \ }}} \ \ \underset{-}{\strut \overset{\text{minus}}{\overbrace{\strut \ -\ }}} \ \ \underset{y_1}{\strut \overset{\text{known y-value}}{\overbrace{\ \strut(-2)\ }}} \ \ \underset{=}{\strut \overset{\text{equals}}{\overbrace{\strut\ =\ }}} \ \ \overset{\text{known slope}}{\overbrace{\strut \underset{m}{\strut \ \ 5\ \ }}} \ \ \underset{(x}{\strut \overset{x}{\overbrace{\strut\ (\ x\ \ }}} \ \ \underset{-}{\strut \overset{\text{minus}}{\overbrace{\strut \ -\ }}} \ \ \underset{x_1)}{\strut \overset{\text{known x-value}}{\overbrace{\strut\ \ 3\ )}}} $$ Then, put it in slope-intercept form by solving for $\,y\,$:

$\,y - (-2) = 5(x - 3)\,$	(start with point-slope form)
$\,y +2 = 5x - 15\,$	(simplify each side)
$\,y = 5x - 17\,$	(subtract $\,2\,$ from both sides)

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Horizontal and Vertical Lines