Beginning Algebra, 11th Edition

274 CHAPTER 4 Systems of Linear Equations and Inequalities

Step 5 State the answer.The club members bought 8 main floor tickets and 22 bal-

cony tickets.

Step 6 Check.The sum of 8 and 22 is 30, so the total number of tickets is correct.

Since 8 tickets were purchased at $148 each and 22 at $65 each, the total of

all the ticket prices is

which agrees with the total amount stated in the problem. NOW TRY

OBJECTIVE 3 Solve problems about mixtures. In Section 2.7,we solved

mixture problems by using one variable. Many mixture problems can also be solved

by using a system of two equations in two variables.

Solving a Mixture Problem Involving Percent

Joe Castillo, a pharmacist, needs 100 L of a 50% alcohol solution. He has on hand a

30% alcohol solution and an 80% alcohol solution, which he can mix. How many

liters of each will be required to make the 100 L of a 50% alcohol solution?

Step 1 Readthe problem. Note the percentage of each solution and of the mixture.

Step 2 Assign variables.

Let the number of liters of 30% alcohol needed,

and y= the number of liters of 80% alcohol needed.

x=

EXAMPLE 3

$148 182 + $65 1222 =$2614,

NOW TRY
EXERCISE 2
General admission rates at a
local water park are $19 for
adults and $16 for children.
If a group of 27 people paid
$462 for admission, how
many adults and how many
children were there?

Summarize the information

in a table. Percents are

written as decimals.

Liters of Percent (as Liters of

Solution a decimal) Pure Alcohol

x 0.30 0.30x

y 0.80 0.80y

100 0.50 0.50 100 1 2

FIGURE 10gives an idea of what is happening in this problem.

Unknown number

of liters, x

Unknown number

of liters, y

100 L of

50% solution

from 80% from 30%

from 30% from 80%

After mixing

+=

FIGURE 10

Step 3 Write two equations.The total number of liters in the final mixture will be

100, which gives the first equation.

To find the amount of pure alcohol in each mixture, multiply the number of

liters by the concentration. The amount of pure alcohol in the 30% solution

added to the amount of pure alcohol in the 80% solution will equal the amount

of pure alcohol in the final 50% solution. This gives the second equation.

0.30x+0.80y= 0.50 11002

x+ y= 100

NOW TRY ANSWER
2.10 adults, 17 children

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