Beginning Algebra, 11th Edition

250 CHAPTER 4 Systems of Linear Equations and Inequalities

OBJECTIVE 3 Solve special systems by graphing. Sometimes the graphs of

the two equations in a system either do not intersect at all or are the same line.

CAUTION With the graphing method, it may not be possible to determine the

exact coordinates of the point that represents the solution, particularly if those coor-

dinates are not integers. The graphing method does, however, show geometrically

how solutions are found and is useful when approximate answers will do.

Solving Special Systems by Graphing

Solve each system by graphing.

(a)

The graphs of these lines are shown in FIGURE 3. The two lines are parallel and

have no points in common. For such a system, there is no solution. We write the

solution set as 0.

2 x+ y= 8

2 x+ y= 2

NOW TRY EXAMPLE 3

EXERCISE 3
Solve each system by
graphing.

(a)

(b)

12 x+ 3 y= 10

4 x+ y= 7

10 x- 6 y= 4

5 x- 3 y= 2

x

y

2 x + y = 8

2 x + y = 2

4

2

0

The lines do

not intersect;

no solution

FIGURE 3

x

y

2 x + 5y = 1

6 x + 15y = 3

3

1

0

Both equations give the

same graph; infinite

number of solutions

FIGURE 4

(b)

The graphs of these two equations are the same line. See FIGURE 4. We can obtain

the second equation by multiplying each side of the first equation by 3. In this case,

every point on the line is a solution of the system, and the solution set contains an in-

finite number of ordered pairs, each of which satisfies both equations of the system.

We write the solution set as

read “the set of ordered pairs such that ” Recall from Section 1.4

that this notation is called set-builder notation.

1 x, y 2 2 x+ 5 y= 1.

51 x, y 2 | 2 x+ 5 y= 16 ,

6 x+ 15 y= 3

2 x+ 5 y= 1

This is the first equation in
the system. See the
Note on the next page.
NOW TRY ANSWERS

3. (a)
   (b) 0

51 x, y 2 | 5 x- 3 y= 26

Solving a Linear System by Graphing

Step 1 Graph each equationof the system on the same coordinate axes.

Step 2 Find the coordinates of the point of intersectionof the graphs if

possible. This is the solution of the system.

Step 3 Checkthe solution in bothof the original equations. Then write the

solution set.

NOW TRY

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