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

y - 2 = 3(x + 4),

the x-intercept is found by setting y=0 and solving for x:

-2 = 3(x + 4)  :  -2 = 3x + 12  :  -14 = 3x  :  x = -14/3.

The y-intercept is found by setting x=0 and solving for y:

y - 2 = 3(4)  :  y - 2 = 12  :  y = 14.

The slope is found by re-writing the equation:

y - 2 = 3x+12

from which we see that the slope is 3.

Finding Equations

METHOD:

  • If the slope m and y-intercept b of a line are known, the equation of the line is y = m x + b.

  • If one point ( x0,y0) on a line and the slope m of a line are known, then the equation of the line is

    y - y0 = m(x - x0).

  • If two points (x0,y0) and (x1,y1) on a line are given, then the line has slope

    m = (y1 - y 0)/ (x1 - x0)

    Its equation is

    y - y0 = m(x - x0).

    with m as shown.

EXAMPLES:

* The equation of the line with slope -2 which passes through the point (3,7) is (y-7) = -2(x - 3).

* 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 point on the graph of f(x) whose x-coordinate is a is ( a , f(a) ).
  • The equation of the line with slope -2 which passes through the point ( a , f(a) ) is (y-f(a)) = -2(x - a).

* The line through (3,4) and (-1,2) has slope m = (2 - 4)/(-1 - 3) = 1/2. It passes through (3,4) so its equation is

y - 4 = (1/2)(x - 3).

It also passes through (-1,2) so its equation is just as well given by y - 2 = (1/2)(x + 1). Both forms can be re-written as y = (1/2)x +(5/2).

Finding Intersections

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

2x - y = -7   and   3 ( y - 1) = x + 4,

re-write these equations as

y = 2 x + 7
y = (1/3) x + (7/3)

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

x + 2y = 12
4x + 8y = -88

are distinct yet parallel (with slope -(1/2)). The slope-intercept form of these equations is

y = -(1/2) x + 6
y = -(1/2) x - 11.

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.

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Version 1
All rights reserved---1996
William A. Bogley
Robby Robson