Lines (§2.3)

The key to understanding a line in the coordinate plane is the slope.

Points and slopes

You'll want to learn this formula: In this formula, you start with two points and calculate the slope:

The slope describes the directions in which you can travel along the line.

Slopes and equations

You'll want to learn this formula: In this formula, you get the equation of a line.

If you don't know the y-intercept, you can still use this equation if you know one of the points; plug it in for x and y, and solve for b. Or use this optional formula:

Conversely, if you have an equation for the line, then solve it for y; you now know what the slope and y-intercept are. It's easy to draw a graph using any point and the slope.

Vertical lines

If a line is vertical, the the run between any two points is zero, so the slope is undefined when you divide by the run. You can also think of this as an infinite slope, since a vertical line is infinitely steep.

If a is the x-coordinate of any point, then the equation for a vertical line is always simply:

Parallel and perpendicular lines

Since the slope of a line indicates its direction, parallel lines always have the same slope. In contrast, perpendicular lines have opposite reciprocal slopes. Also, vertical lines are parallel to one another, and horizontal and vertical lines are each perpendicular to the other.
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