Review of Linear Functions
Finding the Slope and Intercepts
METHOD: Given an equation for a line, the x-intercept is determined by setting y=0, the y-intercept is determined by setting x=0, and the slope is determined by putting the equation into the form y = mx + b. The slope is then m. The slope of the line x=c is undefined or infinite.
EXAMPLE: If the equation of a line is
the x-intercept is found by setting y=0 and solving for x:
The y-intercept is found by setting x=0 and solving for y:
The slope is found by re-writing the equation:
from which we see that the slope is 3.
* The equation of the line with slope -2 which passes through the
point (3,7) is
* Let y=f(x) be a function. Find the equation of the line of slope -2 which intersects the graph of f(x) at the point whose x-coordinate is a:
* The line through (3,4) and (-1,2) has slope m =
METHOD: To find the intersection of two lines, write down both equations in the form y = mx + b, subtract the equations to eliminate y, solve for x, and substitute back into either equation to find y. (If one of the equations is x=c, then x is known so the first step can be skipped.)
EXAMPLE: To find the intersection of the lines given by
re-write these equations as
Then subtract to find that 0 = (5/3) x + (14/3), so x = -14/5. Substitute -14/5 for x in y = 2 x + 7, giving y = 7/5. The point of intersection is thus (-14/5,7/5).
PARALLEL LINES: Two lines do not always meet in a single point. Lines with different slopes always have exactly one point of intersection. Lines with the same slope are either parallel (and never intersect) or identical. If lines have no point of intersection, when you try to solve the simultaneous equations you will end up with a contradictory statement. For example, the two lines
are distinct yet parallel (with slope -(1/2)). The slope-intercept form of these equations is
Subtracting gives 0 = 17, a contradictory statement. This indicates that the lines have no points in common.
If two lines are identical, then both will reduce to the same equation in slope-intercept form.
NOTE: Finding the intersection of two lines is also known as solving a linear system of two equations. You may know any one of several valid methods for doing this.