Intermediate Algebra (11th edition)

Solving a Business Production Problem

A company produces three flat screen television sets: models X, Y, and Z.

• Each model X set requires 2 hr of electronics work, 2 hr of assembly time, and

1 hr of finishing time.

• Each model Y requires 1 hr of electronics work, 3 hr of assembly time, and 1 hr of

finishing time.

• Each model Z requires 3 hr of electronics work, 2 hr of assembly time, and 2 hr of

finishing time.

There are 100 hr available for electronics, 100 hr available for assembly, and 65 hr

available for finishing per week. How many of each model should be produced each

week if all available time must be used?

Step 1 Readthe problem again. There are three unknowns.

Step 2 Assign variables.Then organize the information in a table.

Let the number of model X produced per week,

the number of model Y produced per week,

and z= the number of model Z produced per week.

y=

x=

EXAMPLE 6

240 CHAPTER 4 Systems of Linear Equations

NOW TRY
EXERCISE 6
Katherine has a quilting shop
and makes three kinds of
quilts: the lone star quilt, the
bandana quilt, and the log
cabin quilt.

• Each lone star quilt requires
  8 hr of piecework, 4 hr of
  machine quilting, and 2 hr
  of finishing.


• Each bandana quilt requires
  2 hr of piecework, 2 hr of
  machine quilting, and 2 hr
  of finishing.


• Each log cabin quilt
  requires 10 hr of piecework,
  5 hr of machine quilting,
  and 2 hr of finishing.

Katherine allocates 74 hr for
piecework, 42 hr for machine
quilting, and 24 hr for finish-
ing quilts each month. How
many of each type of quilt
should be made each month
if all available time must be
used?

Each Each Each

Model X Model Y Model Z Totals

Hours of

Electronics Work

2 1 3 100

Hours of

Assembly Time

2 3 2 100

Hours of

Finishing Time

11265

Step 3 Write a system of three equations.The xmodel X sets require 2xhours of

electronics, the ymodel Y sets require 1y(or y) hours of electronics, and the

zmodel Z sets require 3zhours of electronics. Since 100 hr are available for

electronics, we write the first equation.

(1)

Since 100 hr are available for assembly, we can write another equation.

(2)

The fact that 65 hr are available for finishing leads to this equation.

(3)

Notice that by reading across the table, we can easily determine the

coefficients and constants in the equations of the system.

Step 4 Solvethe system of equations (1), (2), and (3).

(1)

(2)

(3)

We find that and

Step 5 State the answer.The company should produce 15 model X, 10 model Y,

and 20 model Z sets per week.

Step 6 Checkthat these values satisfy the conditions of the problem. NOW TRY

x=15,y=10, z= 20.

x+ y+ 2 z= 65

2 x+ 3 y+ 2 z= 100

2 x+ y+ 3 z= 100

x+y+ 2 z= 65

2 x+ 3 y+ 2 z = 100

2 x+ y+ 3 z = 100

NOW TRY ANSWER

6. lone star quilts: 3;
   bandana quilts: 5;
   log cabin quilts: 4