Answer
The answer is: 800 L of the 90% gasoline solution and 400 L of the 75% gasoline solution.
Work Step by Step
We assign the variables:
x = liters of the 90% gasoline mixture
y = liters of the 75% gasoline mixture
Step 1:
Find the equations that represent the problem.
We are mixing both solutions to create a new one,
90% solution + 75% solution = 85% solution -> Eq. 1
We can get the second equation form the wording in the problem.
From " a 90% gasoline mixture",
90% solution $=0.9x$ -> Eq. 2
We can get the third equation form the wording in the problem.
From " a 75% gasoline mixture ",
75% solution $=0.75y$ -> Eq. 3
We can get the fourth equation form the wording in the problem.
From "needed for 1200 L of an 85% gasoline mixture",
85% solution $=0.85(1200)=1020$ -> Eq. 4
Remember that the volume from the new solution has to equal the sum of both volumes of the initial solutions, so
$x+y=1200$
If we solve for x, we get:
$x=1200-y$ -> Eq. 5
Step 2:
Solve the system of equations using the substitution method,
-> Substitute Eq. 2, Eq. 3 and Eq. 4 into Eq. 1
$0.9x+0.75y=1020$ -> Eq. 6
->Substitute Eq. 5 into Eq. 6
$0.9(1200-y)+0.75y=1020$
$1080-0.9y+0.75y=1020$
$-0.15y=1020-1080$
$-0.15y=-60$
$y=\frac{-60}{-0.15}$
$y=400$
-> Substitute the value for $y$ into Eq. 5
$x=1200-400$
$x=800$
Step 3:
The answer is: 800 L of the 90% gasoline solution and 400 L of the 75% gasoline solution.
