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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
78
Number Lines and the Coordinate Plane
Which should you use to graph?
5.1 What characteristic of an equation dictates whether its solution should be
graphed on a number line or a coordinate plane?
A number line is, like any line, one-dimensional, whereas a coordinate plane
is two-dimensional. Equations with one variable are one-dimensional, so their
solutions must be graphed on a number line. Equations with two different
variables have a two-dimensional graph.
5.2 Graph the values w = 6, x = –5, y = , and z = on the number line in
Figure 5-1.
Figure 5-1: Plot w, x, y, and z on this number line.
To plot w, count six units to the right of 0 and mark the value with a solid dot.
Because x is negative, it should be five units to the left of 0. Plot y two-thirds of a
unit to the right of 0 ( of the distance from 0 to 1 on the number line). It
is easier to plot z if you first convert it from am improper fraction into a mixed
number: . It is located 2.25 units to the left of 0, two full units and
then one-fourth of a unit beyond that. All four values, w, x, y, and z, are
illustrated in Figure 5-2.
Figure 5-2: Negative values, like x and z, are located left of the number 0, and positive
values, like w and y, are located to the right of 0.
5.3 Solve the equation 2x – 3(x + 1) – 5 = –6 and graph the solution.
Use the method described in Problems 4.14.36 to isolate x and solve the
equation. Begin by distributing –3 through the parentheses, then combine like
terms, isolate x on the left side of the equation and eliminate the coefficient of x.
A line is
one-dimensional
because it just
allows horizontal
travel, left and right
along the line. On
a coordinate plane,
however, you can
travel horizontally
and vertically, in two
dimensions. Think of an
ant crawling along a
stick as opposed to
crawling around
on a piece of
paper.
If youre
not sure how
to do this, check
out Problems
2.7–2.8.
Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
79
To plot the solution, mark the value –2 on a number line, as illustrated by
Figure 5-3.
Figure 5-3: Only one real number, x = –2, satisfies the equation 2x – 3(x + 1) – 5 = –6.
5.4 Identify lines k and m and point A on the coordinate plane in Figure 5-4.
Figure 5-4: Lines k and m intersect at point A = (0,0).
A coordinate plane is a two-dimensional plane created by two perpendicular
lines. Typically, the horizontal line (m in Figure 5-4) is called the x-axis and
has equation y = 0. The vertical line (k in Figure 5-4) is called the y-axis and
has equation x = 0. The axes intersect at a point called the origin, which has
coordinates (0,0), point A in Figure 5-4.
A point
on the plane
is represented
by coordinates,
sort of like an
address for each
point. Its made up
of two numbers in
parentheses and looks
like B = (2,–1). The
rst number in the
parentheses gives
the horizontal location
of the point (+2 means
two units right of the
origin), and the second
number represents
its vertical location
(1 means one unit
below the origin).
You’ll practice
plotting points
in Problem
5.5.
Chapter Five — Graphing Linear Equations in Two Variables
The Humongous Book of Algebra Problems
80
Note: Problems 5.5–5.6 refer to the points A = (–1,5); B = (4,4); C = (6,–2); D = ;
E = (0,1); and F = (–6,0).
5.5 Plot the points on the coordinate plane in Figure 5-5.
Figure 5-5: Use the coordinates of points A, B, C, D, E, and F to plot the points on this
coordinate plane.
Each coordinate pair has the form (x,y). The left value, x, indicates a signed
horizontal distance from the y-axis. In other words, positive values of x
correspond with points to the right of the y-axis, and negative x-values produce
points left of the y-axis. Similarly, the right value of each coordinate pair, y,
indicates a signed vertical distance from the x-axis. Coordinates with positive
values of y are located above the x-axis, and negative y-values indicate points
below the x-axis. With this in mind, plot each of the points; the results are
graphed in Figure 5-6.

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