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 fertilizer mix.
A) Write a system of linear equations that represents the situation.
B) Solve the system from part (A).
A) To write a system of linear equations that represents the situation, we need to consider the amount of pure fertilizer (in ounces) in each mix and the total volume of the new fertilizer mix (30 fluid ounces).
Let's represent the amount of pure fertilizer in the 20% liquid fertilizer mix as 0.20x (20% of x). Similarly, the amount of pure fertilizer in the 5% liquid fertilizer mix would be 0.05y (5% of y).
Since the gardener combines both mixes to make a final mix of 30 fluid ounces, we can express the following equation based on the total volume:
x + y = 30
The equation for the total amount of pure fertilizer in the mix would be:
0.20x + 0.05y = the total amount of pure fertilizer in the final mix
B) To solve the system of linear equations, we can do either substitution or elimination method. Here, we will solve it using the elimination method.
Let's write the system of equations:
Equation 1: x + y = 30
Equation 2: 0.20x + 0.05y = total amount of pure fertilizer
To eliminate the decimals, we can multiply Equation 2 by 100:
20x + 5y = 100 * (total amount of pure fertilizer)
Now, let's multiply Equation 1 by 5, so it aligns with the coefficients of y:
5x + 5y = 150
Now, we subtract Equation 1 from Equation 2 to eliminate the y variable:
(20x + 5y) - (5x + 5y) = 100 * (total amount of pure fertilizer) - 150
15x = 100 * (total amount of pure fertilizer) - 150 - 5y
Simplifying:
15x = 100 * (total amount of pure fertilizer) - 150 - 5y
As we don't have the value for the total amount of pure fertilizer, we can't find an exact solution in terms of x and y. However, we do have a relationship between x and y through the equation x + y = 30.
To proceed further and obtain numerical values for x and y, we need additional information about the total amount of pure fertilizer.