Solution A is 30% acid and solution B is 60% acid. How many gallons of each solution must be used to create a 60 gallon mixute that is 50% acid. I have to set this up as a system of equations, and I thought one system was x + y = 60, but I can't get an answer, please help
x gallons + y gallons = 60 gallons
y = 60 - x
.3(x) + .6(y) = .5(60)
.3(x) + .6(60-x) = 30
.3x + 36 - .6x = 30
-.3x = -6
x = 20
y = 60 - 20 = 40
check
.3(20) + .6(40) = .5(60)
6 + 24 = 30
To set up a system of equations, let's assign variables to the unknown quantities. Let's use the variables x and y to represent the number of gallons of solution A and solution B, respectively.
Since we need a total of 60 gallons, the first equation is:
x + y = 60
Next, let's consider the acid content in the mixture. Solution A is 30% acid, and you want the final mixture to be 50% acid. For every gallon of solution A, we have 0.3 gallons of acid. Therefore, the acid content from solution A is 0.3x.
Similarly, solution B is 60% acid, so for every gallon of solution B, we have 0.6 gallons of acid. Thus, the acid content from solution B is 0.6y.
Since we want a 60-gallon mixture that is 50% acid, the total acid content of the mixture can be calculated using the equation:
0.3x + 0.6y = 0.5 * 60
This equation can be simplified to:
0.3x + 0.6y = 30
So, the system of equations to solve is:
x + y = 60
0.3x + 0.6y = 30
You can now solve this system of equations to find the values of x and y.
To solve this problem using a system of equations, you need to set up two equations based on the given information.
Let's assume you need to use x gallons of Solution A and y gallons of Solution B to make the 60-gallon mixture.
Equation 1: The total volume of the mixture is 60 gallons, so you can set up the equation:
x + y = 60
Equation 2: The concentration of acid in Solution A is 30% and Solution B is 60%, and you want the resulting mixture to be 50% acid. To express this in terms of acid, you need to calculate the total amount of acid in each solution and equate it to the total amount of acid in the mixture. The equation will be based on the principle that the amount of acid is proportional to the volume of the solution.
The acid in Solution A is 30% of the total volume x:
0.3x
The acid in Solution B is 60% of the total volume y:
0.6y
The acid in the resulting mixture is 50% of the total volume of 60 gallons:
0.5 * 60 = 30
Thus, the equation representing the acid content is:
0.3x + 0.6y = 30
Now you have a system of equations:
Equation 1: x + y = 60
Equation 2: 0.3x + 0.6y = 30
You can solve this system of equations using various methods, such as substitution or elimination. Using the elimination method, multiply the first equation by 0.3 to match the coefficient of x in the second equation:
0.3(x + y) = 0.3 * 60
0.3x + 0.3y = 18
Now you have the following system of equations:
Equation 1: 0.3x + 0.3y = 18
Equation 2: 0.3x + 0.6y = 30
Subtract Equation 1 from Equation 2 to eliminate the variable x:
(0.3x + 0.6y) - (0.3x + 0.3y) = 30 - 18
0.6y - 0.3y = 12
0.3y = 12
y = 12 / 0.3
y = 40
Now, substitute the value of y back into Equation 1:
x + 40 = 60
x = 60 - 40
x = 20
Therefore, you need to use 20 gallons of Solution A and 40 gallons of Solution B to create a 60-gallon mixture that is 50% acid.