Let \( x \) be the cost per pound of trail mix and \( y \) be the cost per pound of jelly beans. We can set up the following system of equations based on the information provided:
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For 8 pounds of trail mix and 3 pounds of jelly beans costing $29: \[ 8x + 3y = 29 \]
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For 2 pounds of trail mix and 5 pounds of jelly beans costing $20: \[ 2x + 5y = 20 \]
Now we will solve this system of equations.
Step 1: Solve the second equation for \( x \)
From the second equation: \[ 2x + 5y = 20 \]
We can express \( x \) in terms of \( y \): \[ 2x = 20 - 5y \] \[ x = 10 - \frac{5}{2}y \]
Step 2: Substitute \( x \) into the first equation
Now, we plug this expression for \( x \) into the first equation: \[ 8(10 - \frac{5}{2}y) + 3y = 29 \]
Step 3: Distribute and simplify
Distributing \( 8 \): \[ 80 - 20y + 3y = 29 \]
Combining like terms: \[ 80 - 17y = 29 \]
Step 4: Solve for \( y \)
Subtract \( 80 \) from both sides: \[ -17y = 29 - 80 \] \[ -17y = -51 \]
Now divide by \( -17 \): \[ y = 3 \]
Step 5: Substitute \( y \) back to find \( x \)
Now, substitute \( y = 3 \) back into the expression for \( x \): \[ x = 10 - \frac{5}{2}(3) \] \[ x = 10 - \frac{15}{2} \] \[ x = 10 - 7.5 \] \[ x = 2.5 \]
Conclusion
The costs are:
- The cost per pound of trail mix \( x \) is $2.50.
- The cost per pound of jelly beans \( y \) is $3.00.