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 4.1–4.36 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 you’re

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. It’s 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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