Let's analyze the given equations:
- The equation \(4x + 5y = 30.50\) represents the purchase of 4 pounds of meat and 5 pounds of cheese for a total of $30.50.
- The equation \(11x + 14y = 84.50\) represents the purchase of 11 pounds of meat and 14 pounds of cheese for a total of $84.50.
In these equations:
- The variable \(x\) represents the price per pound of meat.
- The variable \(y\) represents the price per pound of cheese.
To find the prices, we can solve the system of equations.
We can use the method of substitution or elimination. Here, we'll use elimination for convenience.
First, let's adjust the first equation (multiply by 2.75) so that the coefficient of \(y\) matches in both equations:
-
Multiply the first equation by \(2.75\): \[ 2.75(4x + 5y) = 2.75 \cdot 30.50 \] \[ 11x + 13.75y = 83.875 \]
-
Now we have the adjusted first equation \(11x + 13.75y = 83.875\) and the second equation \(11x + 14y = 84.50\).
Now we can subtract the adjusted equation from the second equation to eliminate \(x\): \[ (11x + 14y) - (11x + 13.75y) = 84.50 - 83.875 \] \[ 0x + 0.25y = 0.625 \] \[ 0.25y = 0.625 \] Now, divide both sides by 0.25: \[ y = \frac{0.625}{0.25} = 2.5 \] So, the price per pound of cheese, \(y\), is $2.50.
Now substitute \(y = 2.5\) back into one of the original equations to find \(x\): Using the first equation: \[ 4x + 5(2.5) = 30.50 \] \[ 4x + 12.5 = 30.50 \] Subtract 12.5 from both sides: \[ 4x = 30.50 - 12.5 \] \[ 4x = 18.00 \] Now divide by 4: \[ x = \frac{18.00}{4} = 4.50 \] So, the price per pound of meat, \(x\), is $4.50.
In conclusion:
- The variable \(x\) represents the price per pound of meat.
- The variable \(y\) represents the price per pound of cheese.
- The deli charges $4.50 per pound of meat and $2.50 per pound of cheese.
9 answers