Slope Intercept Form of a Line

Every linear equation can be represented by a line (thus, the name linear equation!).  The equations take the form of $y=mx+b$.  (There are other ways to write linear equations, but $y=mx+b$ is the easiest form for graphing and finding the graphs of a line.   

$y=mx+b$ has several components:

• $x$ and $y$:
  • These are coordinates ($x$, $y$).  $x$ and $y$ represent the ($x$,$y$) coordinates of every point that is on the line represented by the equation (and remember, lines are infinite so there are infinite correct ($x$,$y$) coordinates).
• $m$
  • Slope: $m$ represents the slant of a line, also expressed as rise (upward motion) over run (horizontal motion), or change in $y$ over change in $x$.  A positive $m$ rises from left to right.  A negative $m$ drops from right to left.
• $b$: 
  • Y-intercept: The point where the line crosses the y axis. 

There area few other terms that are helpful to know when graphing lines:

• Origin: The point (0,0) where the x and y axes cross.
• x-axis: The horizontal the in the graph.
• y-axis: The vertical line in the graph.

How can you find the equation of a line using a graph?

Start with the equation $y = mx + b$, where $x$ and $y$ represent coordinates of the points on the line,  $m$=slope, $b$= the y-intercept.

To find the equation of a line, the easiest place to start is with the y-intercept. 

Look at the y-axis, and find the point where the line crosses the axis.  

In this case, at point 2.  Substitute the y-intercept (in this case, 2) for b in your equation.            

$y = mx +\mathbf{b}$

$y = mx +\mathbf{ 2}$

Next, find the slope of the line.  Find two points on the line where it’s easy to figure out the coordinates.  In this case the y-intercept

(0,2) and the x-intercept (5,0) are the easiest points to use.  Starting from one point (0,2) count how many spaces you need go

vertically (-2), and horizontally (5) to get to the other point (5,0).  Those two go into a fraction (rise over run) to make your slope.

$\text{slope}=\dfrac{\text{rise}}{\text{run}}=\dfrac{-2}{5}$

Then, plut the slope that you just found into your equation ($m$):

$y=\dfrac{-2}{5}x+2$

For any line, you should be able to plug the slope ($m$), the y-intercept ($b$), or the coordinates ($x$,$y$) from any point on the line into the equation $y=mx+b$ to find any missing variables and write the equation of the line.

Note: For any line, you should be able to plug in the (x,y) coordinates of any point on the line into the equation and the equation will come out true.  You should be able to plug in the coordinates from any point not on the line and the equation should come out false.

Note: When working with parallel lines, remember, the slopes are the same ($m=m$).

Note: when working with perpendicular lines, remember, the slopes are the opposite inverse ($m=-\dfrac{1}{m}$)