Chapter Five — Graphing Linear Equations in Two Variables

The Humongous Book of Algebra Problems

83

Figure 5-9: The graph of y = 2 is a horizontal line two units above the x-axis.

Graphing with a Table of Values

Plug in some x’s, plot some points, call it a day

Note: Problems 5.9–5.11 refer to the linear equation 2x – y = 4.

5.9 How many coordinates are required to draw the graph of the equation?

Geometric principles dictate that two points on the same plane are required

to draw the line that contains those points. If, however, you use coordinates to

draw a linear graph, identifying and plotting at least one additional point is

advised. The third point serves as a quick way to check your work. If it lies on

the same line as the other two points, you’re far less likely to have made an error

in your calculations.

Note: Problems 5.9–5.11 refer to the linear equation 2x – y = 4.

5.10 Use a table of values to identify three points on the graph of the line.

A table of values is a brute force arithmetic technique that generates lists of

coordinate pairs satisfying the given equation. After a sufﬁcient number of

points has been identiﬁed and plotted, all that remains is to “connect the dots”

on the coordinate plane to graph the equation. The number of points necessary

to create an accurate graph varies based on the complexity of the equation.

I call the

third point a

“check point.” If

you make a mistake

nding either of the

other two points, the

check point won’t be

on the graph. That’s

a red ag to go

back and check

your arithmetic

for all three

coordinates.

Chapter Five — Graphing Linear Equations in Two Variables

The Humongous Book of Algebra Problems

84

Because 2x – y = 4 is a linear equation in two variables, a minimum of two points

is required to graph it. However, as explained in Problem 5.9, three coordinates

should be plotted to better ensure your calculations are correct.

Begin by solving the equation for y.

Construct a table with columns labeled, from left to right, “x,” “y = 2x – 4,” and

“y.” The outside columns, x and y, will eventually house the coordinates you will

plot on the graph. The inner column contains the equation just solved for y.

Choose three values of x to plug into the equation y = 2x – 4. Select x-values that

do not produce large or unnecessarily complicated results. Write the three x-

values you chose in the left column of the table.

Substitute each x-value into the equation to determine the corresponding values

of y and record those in the right column.

Combine the x- and y-values in each row to conclude that the graph contains

the points (–1,–6), (0,–4), and (1,–2).

The ab-

solute value

graphs at the end

of the chapter are

a little trickier. You

could get away with

using only two points

to make the graph,

but you’d be showing

off. Better to use

more points as the

equations get more

complicated.

See Problems 4.37–

4.43 if you need help

with this step.

Whenever

possible, choose

simple, small numbers

like x = –1, x = 0,

and x = 1.

Chapter Five — Graphing Linear Equations in Two Variables

The Humongous Book of Algebra Problems

85

Note: Problems 5.9–5.11 refer to the linear equation 2x – y = 4.

5.11 Graph the linear equation using the table of values generated by Problem 5.10.

Plot the points identiﬁed in Problem 5.10 and connect them to create the linear

graph illustrated by Figure 5-10.

–4

Figure 5-9: The graph of 2x – y = 4 passes through the points (–1,–6), (0,–4), and

(1,–2).

Note: Problems 5.12–5.13 refer to the linear equation x + 3y = –2.

5.12 Use a table of values to identify three points on the graph of the equation.

Solve the equation for y.

Create a table of values based on three simple values of x, as demonstrated in

Problem 5.10.

This time

the table of

values does not

contain x = –1 and

x = 0. If you plug in x =

1, however, the fraction

turns into an integer.

That’s why the other

two values, x = –2 and

4, are used as well—

they eliminate the

fraction.

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