Let's define the variables:

• Let $x$ represent the weight of a large box (in pounds).
• Let $y$ represent the weight of a small box (in pounds).

We are given the following information:

1. The combined weight of a large box and a small box is 80 pounds, so: $x + y = 80 \quad \text{(Equation 1)}$

2. The truck is transporting 60 large boxes and 70 small boxes, and the total weight is 5100 pounds, so: $60x + 70y = 5100 \quad \text{(Equation 2)}$

Now, we have a system of two equations: $x + y = 80 \quad \text{(Equation 1)}$ $60x + 70y = 5100 \quad \text{(Equation 2)}$

Step 1: Solve for $y$ in terms of $x$ from Equation 1.

From Equation 1: $y = 80 - x$

Step 2: Substitute this expression for $y$ into Equation 2.

Substitute $y = 80 - x$ into $60x + 70y = 5100$: $60x + 70(80 - x) = 5100$ Now simplify the equation: $60x + 5600 - 70x = 5100$ Combine like terms: $-10x + 5600 = 5100$ Solve for $x$: $-10x = 5100 - 5600$ $-10x = -500$ $x = \frac{-500}{-10} = 50$

Step 3: Solve for $y$.

Now that we know $x = 50$, substitute this back into $y = 80 - x$: $y = 80 - 50 = 30$

Final Answer:

• The weight of a large box is 50 pounds.
• The weight of a small box is 30 pounds.

Would you like a more detailed explanation or help with anything else?

Here are some related questions you might find interesting:

1. How would the solution change if the number of large boxes was doubled?
2. What would happen to the total weight if the weight of the small boxes increased by 10 pounds?
3. How can the same method be used to solve a similar problem with more types of boxes?
4. How could this problem be solved using matrix methods?
5. What would the system of equations look like if the truck carried 100 large boxes and 50 small boxes instead?

Tip: When dealing with word problems like this, defining variables early on is crucial to simplifying the system of equations.