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:


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

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Perpendicular Lines

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