A gardener combines x fluid ounces of a 20% liquid fertilizer and 80% water mix with y fluid ounces of a 5% liquid fertilizer and 95% water mix to make 30 fluid ounces of a new 10% fertilizer mix.
A) Write a system of linear equations that represents the situation.
B) Solve the system from part (A). Make sure you show your work.
This is what question 15 should say. It was missing the "10%" before
obvious first equation: x + y = 30
amount of liquid fertilizer we have :
.2x + .05y = .1(30)
by 100:
20x + 5y = 300
4x + y = 60
so we have:
x + y = 30
4x + y = 60
continue from here,
looks straight forward from here, comes out nicely
A) To represent the given situation as a system of linear equations, we need to consider the amount of pure fertilizer (i.e., the active ingredient) in each mix. Let's assume that the 20% liquid fertilizer contains x1 ounces of the pure fertilizer, and the 5% liquid fertilizer contains x2 ounces of the pure fertilizer.
The total amount of pure fertilizer in the 20% mix would be 20% of x fluid ounces, which can be given as: 0.20x1.
Similarly, the total amount of pure fertilizer in the 5% mix would be 5% of y fluid ounces, which can be given as: 0.05x2.
Since the gardener combines x fluid ounces of the 20% fertilizer mix and y fluid ounces of the 5% fertilizer mix to make a 30 fluid ounces of a 10% fertilizer mix, we can write the following equations:
Equation 1: x1 + x2 = 30 (The total amount of fluid ounces in the new mix)
Equation 2: 0.20x1 + 0.05x2 = 0.10(30) (The total amount of pure fertilizer in the new mix)
B) Let's solve the system of linear equations to find the values of x1 and x2.
Using equation 1:
x1 + x2 = 30
Solving equation 2 for x2:
0.20x1 + 0.05x2 = 0.10(30)
0.20x1 + 0.05x2 = 3
Multiplying equation 2 by 20 to eliminate decimals:
4x1 + x2 = 60
Now we have the following system of equations:
x1 + x2 = 30
4x1 + x2 = 60
Multiplying equation 1 by -1 to eliminate x1:
-x1 - x2 = -30
Adding the modified equation 1 to equation 2:
4x1 + x2 -x1 - x2 = 60 - 30
3x1 = 30
x1 = 10
Substituting the value of x1 into equation 1:
10 + x2 = 30
x2 = 20
Therefore, the solution to the system of linear equations is x1 = 10 and x2 = 20.
Hence, the gardener needs 10 fluid ounces of the 20% liquid fertilizer and 20 fluid ounces of the 5% liquid fertilizer to make a 30 fluid ounces of a new 10% fertilizer mix.