To determine at which job Percy earns the greater hourly wage, we need to solve the system of equations given:
- \( 3x + 2y = 36.50 \) (Equation 1)
- \( 2x + 5y = 50.00 \) (Equation 2)
We can solve this system using substitution or elimination. Here, we'll use the elimination method.
First, let's manipulate Equation 1 to eliminate one of the variables. We can multiply Equation 1 by 5 and Equation 2 by 2 to align the coefficients of \(y\):
\[ 5(3x + 2y) = 5(36.50) \implies 15x + 10y = 182.50 \quad (Equation 3) \]
\[ 2(2x + 5y) = 2(50.00) \implies 4x + 10y = 100.00 \quad (Equation 4) \]
Now we subtract Equation 4 from Equation 3:
\[ (15x + 10y) - (4x + 10y) = 182.50 - 100.00 \]
This simplifies to:
\[ 11x = 82.50 \]
Now divide both sides by 11:
\[ x = \frac{82.50}{11} = 7.50 \]
Now that we have \(x\) (the hourly wage at the library), we need to find \(y\) (the hourly wage at the coffee cart). We can substitute \(x\) back into either Equation 1 or Equation 2. We will use Equation 1:
\[ 3(7.50) + 2y = 36.50 \]
This simplifies to:
\[ 22.50 + 2y = 36.50 \]
Subtract 22.50 from both sides:
\[ 2y = 14.00 \]
Now divide by 2:
\[ y = 7.00 \]
So, we found \(x = 7.50\) (library) and \(y = 7.00\) (coffee cart).
Percy earns a greater hourly wage of $7.50 at the library.
Therefore, the answer is: Percy earns a greater hourly wage of $7.50 at the library.