PointSlope Form
Want some other practice with 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 pointslope 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: $$\cssId{s15}{m = \frac{yy_1}{xx_1}}$$ or, equivalently, $$\cssId{s17}{y  y_1 = m(x  x_1)}$$
This gives us an extremely useful equation of a line, as summarized below:
Since this equation is ideally suited to the situation where you know a point and a slope, it is appropriately called pointslope form.
Important Things to Know About PointSlope 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 infinitelymany 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 slopeintercept 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 pointslope form.

Remember—just as expressions have
lots of different names, so do sentences.
Every nonvertical line can be written
in any of these forms:
 pointslope form: $$\cssId{s66}{y  y_1 = m(x  x_1)}$$
 slopeintercept 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!)

pointslope form: $$y  2 = 5(x  3)$$

slopeintercept form: $$y = 5x 13$$

general form: $$5x  y  13 = 0$$

Example
$y$  minus  known $y$value 
equals  known slope 
$($  $x$  minus  known $x$value 
$)$ 
$y$  $$  $(2)$  $=$  $5$  $($  $x$  $$  $3$  $)$ 
$y$  $$  $y_1$  $=$  $m$  $($  $x$  $$  $x_1$  $)$ 
Then, put it in slopeintercept form by solving for $\,y\,$:
$y  (2) = 5(x  3)$  start with pointslope form 
$y +2 = 5x  15$  simplify each side 
$y = 5x  17$  subtract $\,2\,$ from both sides 