Parallel and Perpendicular Lines
Parallel lines are lines that don’t intersect. In coordinate geometry, they can also be described as lines with the same slope.
Perpendicular lines are lines that intersect at a right angle. In coordinate geometry, perpendicular lines have opposite, reciprocal slopes. That is, a line with slope m is perpendicular to a line with a slope of –1⁄m.
In the figure, lines q and r both have a slope of 2, so they are parallel. Line s is perpendicular to both lines q and r, and thus has a slope of –1⁄2.
Equation of a Line
A line in coordinate geometry can be described by an equation containing the variables x and y. For the Math IC, you need to understand thoroughly two forms of the equation of a line: the slope-intercept form and the point-slope form.
Slope-Intercept Form
The slope-intercept form of the equation of a line is:
where m is the slope of the line, and b is the y-intercept of the line. Both are constants.
The y-intercept of a line is the y-coordinate of the point where the line intersects the y-axis. Likewise, the x-intercept of a line is the x-coordinate of the point where the line intersects the x-axis. Therefore, if given the slope-intercept form of the equation of a line, you can find both intercepts.
For example, in order to find the y-intercept, simply set x = 0 and solve for the value of y. For the x-intercept, set y = 0 and solve for x.
To sketch a line given in slope-intercept form, first plot the y-intercept, and then use the slope of the line to plot another point. Connect the two points to form your line. In the figure below, the line y = –2x + 3 is graphed.
Since the slope is equal to –2, the line descends 2 units for every 1 unit it moves in the positive x direction. The y-intercept is at 3, so the line crosses the y-axis at (0, 3). For practice, solve for the x-intercept.
Point-Slope Form
The point-slope form of the equation of a line is:
where m is the slope of the line, and (x1, y1) is a point on the line.
The point-slope form and slope-intercept form are alternative ways of expressing the same equation. In fact, the slope-intercept form is the point-slope form taken at the y-intercept, or the point (0, y1):
Since y1 = b (the y-intercept is simply the y-coordinate of the point at which x = 0), the two forms are equal.
The slope-intercept form of the line equation is the more common of the two, but the point-slope form is useful when your only is the slope and a point (hence the name point-slope form).
