The problem in the image reads:

Amy owns an organic nut shop. A customer buys 5 pounds of walnuts and 3 pounds of pecans for $21. Another customer buys 4 pounds of walnuts and 8 pounds of pecans for $28.50. What is the cost of one pound of pecans?

Step-by-step solution:

We need to set up a system of equations to solve for the prices of walnuts and pecans.

Let:

• $w$ be the price per pound of walnuts.
• $p$ be the price per pound of pecans.

From the two transactions:

1. $5w + 3p = 21$ (First customer buys 5 pounds of walnuts and 3 pounds of pecans for $21)
2. $4w + 8p = 28.50$ (Second customer buys 4 pounds of walnuts and 8 pounds of pecans for $28.50)

Solving the system of equations:

Step 1: Multiply the first equation by 8 and the second equation by 3 to align the coefficients of $p$:

$8(5w + 3p) = 8(21) \quad \Rightarrow \quad 40w + 24p = 168$ $3(4w + 8p) = 3(28.50) \quad \Rightarrow \quad 12w + 24p = 85.50$

Step 2: Subtract the second equation from the first:

$(40w + 24p) - (12w + 24p) = 168 - 85.50$ $28w = 82.50$ $w = \frac{82.50}{28} \quad \Rightarrow \quad w = 2.95$

So, the price of one pound of walnuts is $2.95.

Step 3: Substitute $w = 2.95$ into the first equation to solve for $p$:

$5(2.95) + 3p = 21$ $14.75 + 3p = 21$ $3p = 21 - 14.75 \quad \Rightarrow \quad 3p = 6.25$ $p = \frac{6.25}{3} \quad \Rightarrow \quad p = 2.08$

So, the price of one pound of pecans is $2.08.

Would you like more details or explanations on any part of the solution? Here are five related questions you can consider:

1. How would the system of equations change if the prices of walnuts and pecans were different?
2. What would happen if a third transaction was introduced? Could we solve for both prices without a system of equations?
3. How can this method be applied to other two-variable cost problems?
4. Can we solve this system of equations using a matrix method, such as Gaussian elimination?
5. How does the substitution method compare to the elimination method in terms of efficiency?

Tip: When solving systems of equations, it can be helpful to align coefficients to simplify the elimination process.