Two POINTS make a LINE

We can connect two points with a straight line.
Every straight line can be represented by an equation: y = mx + b.

The coordinates of every point on the line will solve the equation if you substitute them in the equation for x and y.
The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept.
The y-intercept of this line is the value of y at the point where the line crosses the y axis.

To graph the equation of a line, we plot at least two points whose coordinates satisfy the equation, and then connect the points with a line. We call these equations "linear" because the graph of these equations is a straight line.
There are two important things that can help you graph an equation, slope and y-intercept.

Slope

We're familiar with the word "slope" as it relates to mountains. Skiers and snowboarders refer to "hitting the slopes." On the coordinate plane, the steepness, or slant, of a line is called the slope. Slope is the RATIO of the change in the y-value over the change in the x-value. Carpenters and builders call this ratio the "rise over the run." Using any two points on a line, you can calculate its slope using this formula.


Let's use these two points to calculate the slope m of this line.
A = (1,1) and B = (2,3)


Subtract the y value of point A from the y-value of point B to find the change in the y value (2 — 1), which is 2. Then subtract the x value of point A from the x value of point B to find the change in x (3 — 1), which is 1. The slope is 2 divided by 1, or 2.

When a line has positive slope, like this one, it rises from left to right.
When a line has positive slope, like this one, it rises from right to left.
Let's look at another line. This line has a negative slope, it falls from left to right. We can take any two points on this line and find the slope. Let's take C (0, -1) and D (2, -5).
Using these two points, we can calculate the slope of this line. We subtract the y value of point C from the y value of point D, and the x value of point C from the x value of point D, and divide the first value by the second value. The slope is -2.

Y-Intercept

There's another important value associated with graphing a line on the coordinate plane. It's called the "y intercept" and it's the y value of the point where the line intersects the y- axis. For this line, the y-intercept is
"negative 1." You can find the y-intercept by looking at the graph and seeing which point crosses the y axis. This point will always have an x coordinate of zero. This is another way to find the y-intercept, if you know the equation, the y-intercept is the solution to the equation when x = 0.

Equations

Knowing how to find the slope and the y-intercept helps us to graph a line when we know its equation, and also helps us to find the equation of a line when we have its graph. The equation of a line can always be written in this form, where m is the slope and b is the y-intercept:
y = mx + b


Why is m used as the variable name for slope?