# Algebra: Finding the Equation of a Line

On this page we hope to clear up problems that you might have with finding the equation of a line.

Many times, you'll have the graph of an equation shown to you and you'll need to find the equation.  This seems a very daunting task, but it's actually quite easy!

For example, take a horizontal line such as y = 2.  Every point on that graph is 2 units above teh x axis.  All horizontal lines have equations that are written in the same format, such as y = -4.5.  Because they're all written the same way, we can come up with a general formula for horizontal lines.  It is the following equation (where k represents any real number): y = k.  Under the same assumptions, the general formula for vertical lines can be written as follows (where k represents any real number): x = k.

Linear equations such as y = .009x + 34 also have a general equation that can represent any linear equation.  It is written as follows: y = mx + b.

The things to remember about the above formula, which is called the slope-intercept formula, are outlined below.

1.  Since you know a line with an equation is that form cannot be horizontal or vertical, all you need to find are m and b to find the equation.
2.  b is called the intercept.  It is the point when the line crosses the y axis.
3.  m is called the slope.  The slope has both a sign (either + or -) and a value (the number behind the sign).  For example, the equation y = -3x + 4 has a negative sign and a value of three.  Therefore, the slope is -3.  When looking at a graph, you can always tell if the slope is negative or positive by the direction it points.  When looking for the sign of a slope, look at the left side of the graph.  Then, look at the right side of the graph.  If the right side is lower than the left side, the line has a negative slope, if the right side is higher, the line has a positive slope.

Example:

```              The graph: Example Graph
```
To find the value of the slope, you compute the rise over the run.  To do that, pick two points on the line at random and then draw a line through each of those points that runs parallel to the coordinate axes.  Count the number of units between the point on the line and where the two additional lines you drew intersect.  The number of units on the horizontal line is the run and the number of units on the vertical line is the rise.  Dividing the rise by the run gives you the value of the slope.

Example:

```              The graph: Example Graph
```
Example:
```1. Find the equation of the line graphed in the accompanying figure.
Solution: The desired equation is in the slope-intercept form.
You need to find m and b.

By looking at the graph, you can see that b must be 3.
Also, by inspecting the graph, you can see that the slope is
negative because the left end of the line is higher than the
right end.

Now you need to find the value of the slope.  Pick any two
points on the line and draw a line through each point that
is horizontal to the coordinate axes.  (For simplicity's sake,
we will use b, which is (0,3) and the point given in
the problem, (2,0).)  Count the number of units on the
vertical line and horizontal line.  The vertical line is the
rise and the horizontal line is the run.  Compute
the slope's value by putting the rise over the run.  In this
case, the rise is 3 and the run is 2.
Therefore, the value of the slope is (3/2).  Combine the
sign of the slope and the value of the slope to get the
complete number for m.  It is -(3/2).

Plug the values you found into the slope-intercept
general formula and you get the following:
3
y = - -x + 3
2
```