# The care and feeding of lines and linear equations

## Slope

When discussing lines in the plane, it is very useful to know
how "steep" the line is. Mathematically, we measure steepness
using a quantity called * slope.* If you have two
different points
(a,b) and (c,d) on the line, then the slope of the line, usually
denoted m, is defined by
m=(d-b)/(c-a)

It turns
out that the answer remains the same no matter what two distinct
points we choose on the line, so we can say that the slope of
the line is this quantity. Many people remember this calculation
as "the rise over the run" as you compute it by dividing the
amount the line "rises" between the two points by the amount
the line "runs".
Some lines "slope upward" and others "slope downward". We can
tell which of these is happening by looking at whether the
slope is positive or negative. If a line has positive slope,
then the y-coordinate increases as the x-coordinate increases,
so the line "slopes upward." If a line has negative slope,
then the y-coordinate decreases as the x-coordinate increases,
so the line "slopes downward."

Of course, some lines lay "flat", that is, they neither rise
nor fall. This fact is reflected in a slope of zero.

There is one other possibility: The line may be vertical.
In this case, the x-coordinate is not changing and so when we
compute the slope, we end up dividing by zero, which is illegal
in mathematics. Therefore, we say that the slope of a
vertical line is undefined.

In summary:

- A positive slope, m>0, indicates that the line "slopes upward".
- A negative slope, m<0, indicates that the line "slopes downward".
- A slope of zero, m=0, indicates that the line is horizontal.
- An undefined slope indicates that the line is vertical.

## Intercepts

The * x-intercept * of a line is the point where the line
crosses the x-axis, and
the * y-intercept * of a line is the point where the line
crosses the y-axis.
We'll focus our attention on the y-intercept as it is used more
often. The thing to remember is that the x-coordinate of the
y-intercept is zero, so
the y-intercept is a point of the form (0,b).

## Equations of lines

Every non-vertical line in the plane has an equation of the form

y = mx + b

This form, known as * slope-intercept form *, allows us
to read the slope m and the y-intercept (0,b) directly from the equation.
For instance, given the line
y = 4x - 7

we can tell at once that this line has slope m = 4 and y-intercept
at (0,-7).
Thus, we know from the equation that this line slopes upward
and hits the y-axis below the origin.
For vertical lines, the generic equation is

x = a

Notice that the variable y fails to appear in the equation of a vertical
line.

## Intersections of Lines

TEXT TO BE PROVIDED

## Perpendicular Lines

TEXT TO BE PROVIDED

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