Final answer:
The question is about using a system of equations to determine the amount the actor invested at two different interest rates. After establishing the relationships among the invested amounts and the interest earned, the solution involves substituting one equation into another and then solving for the variables to find the exact values of the investments.
Step-by-step explanation:
To solve the question of how much the actor invested at each rate, we will use a system of equations as the actor invests some money at 8% and $50,000 more than four times that amount at 11%. We'll define the amount invested at 8% as x and the amount invested at 11% as y.
Step 1: Establish the equations
1) x + y = total investment
2) 0.08x + 0.11y = total interest of $74,140
From the given information, we have an additional relationship between x and y: y = 4x + 50,000. Let's substitute this into our equations.
Step 2: Substitute and solve
Replacing y with 4x + 50,000 in the second equation, we get:
0.08x + 0.11(4x + 50,000) = 74,140
Solving for x, we find the investment at 8% and then use it to calculate the investment at 11%.
Step 3: Verify the solution
Insert the values of x and y back into the original equations to confirm that they satisfy the condition that the total interest earned is $74,140.