SECTION 4.4 Applications of Linear Systems^275
Be sure to write
twoequations.These two equations form this system.
(1)
(2)Step 4 Solvethe system by the substitution method. Solving equation (1) for x
gives Substitute for xin equation (2).
(2)
Let
Distributive property
Combine like terms.
Subtract 30.
Divide by 0.50.Then
Step 5 State the answer.The pharmacist should use 60 L of the 30% solution and
40 L of the 80% solution.
Step 6 Check.Since and this mixture
will give 100 L of the 50% solution, as required. NOW TRY
60 + 40 = 100 0.30 1602 +0.80 1402 =50,
x= 100 - y= 100 - 40 =60.
y= 40
0.50y= 20
30 +0.50y= 50
30 - 0.30y+0.80y= 50
0.30 1100 - y 2 +0.80y= 50 x= 100 - y.
0.30x+0.80y= 50
x= 100 - y. 100 - y
0.30x+ 0.80y= 50 0.50 11002 = 50
x+ y= 100
Distribute 0.30 to
both100 and -y.NOW TRY
EXERCISE 3
A biologist needs 80 L of a
30% saline solution. He has a
10% saline solution and a 35%
saline solution with which to
work. How many liters of each
will be required to make the
80 L of a 30% solution?
NOW TRY ANSWER
3.16 L of 10%; 64 L of 35%
NOTE In Example 3,we could have used the elimination method. Also, we could
have cleared decimals by multiplying each side of equation (2) by 10.
OBJECTIVE 4 Solve problems about distance, rate (or speed), and time.
Problems that use the distance formula were solved in Section 2.7.
Solving a Problem about Distance, Rate, and TimeTwo executives in cities 400 mi apart drive to a business meeting at a location on the
line between their cities. They meet after 4 hr. Find the rate (speed) of each car if one
travels 20 mph faster than the other.
Step 1 Readthe problem carefully.
Step 2 Assign variables.
Let the rate of the faster car,
and the rate of the slower car.
Make a table and draw a sketch. See FIGURE 11.
y=
x=
EXAMPLE 4
d=rt
Since each car travels for
4 hr, the time tfor each car
is 4. Find d, using d=rt.rt d
Faster Car x 4 4 x
Slower Car y 4 4 yCars meet after 4 hr.4 x 400 mi 4 yFIGURE 11