a fruit company delivers its fruit in two types of boxes: large and small. A delivery of 2 large boxes and 3 small boxes has a total weight of 47 kilograms. A delivery of 6 large boxes and 5 small boxes has a total weight of 115 kilograms. How much does each type of box weigh?
 


Let's denote the weight of a large box by \( L \) (in kilograms) and the weight of a small box by \( S \) (in kilograms). We can set up the following system of linear equations based on the information given:

\[
\begin{align*}
2L + 3S &= 47 \quad \text{(Equation 1)} \\
6L + 5S &= 115 \quad \text{(Equation 2)}
\end{align*}
\]

### Step 1: Solve Equation 1 for one variable

Let's solve Equation 1 for \( L \):

\[
2L = 47 - 3S
\]

\[
L = \frac{47 - 3S}{2}
\]

### Step 2: Substitute into Equation 2

Now, substitute \( L \) from the first equation into the second equation:

\[
6\left(\frac{47 - 3S}{2}\right) + 5S = 115
\]

Multiply through by 2 to eliminate the fraction:

\[
3(47 - 3S) + 10S = 230
\]

\[
141 - 9S + 10S = 230
\]

\[
141 + S = 230
\]

### Step 3: Solve for \( S \)

\[
S = 230 - 141
\]

\[
S = 89
\]

### Step 4: Substitute \( S \) back into the equation for \( L \)

Substitute \( S = 89 \) back into the expression for \( L \):

\[
L = \frac{47 - 3(89)}{2}
\]

\[
L = \frac{47 - 267}{2}
\]

\[
L = \frac{-220}{2} = -110
\]

This result doesn't seem physically plausible, indicating a calculation mistake or misunderstanding in assumptions. Let's re-evaluate starting directly from the system.

### Re-evaluation:
#### Original Equations:
\[
2L + 3S = 47 
\]
\[
6L + 5S = 115 
\]
#### Simplifying Equation 1 by isolating:
Multiply the entire Eq1 by 3 to align with Eq2.
\[
6L + 9S = 141
\]
\[
6L + 5S = 115
\]

### Correct Approach:
Subtract Eq2 from above:
\[  4S = 26 
S = 6.5 
\]
Substitute \( S= 6.5 \) back:
\[ 2L + 3(6.5) = 47 
2L + 19.5 = 47 
2L= 27.5 \]

### Final Result:
\[ S = 6.5 kg , L= 13.75 kg]

Would you like further details?  

---

1. What if the ratio of box types were different?
2. Can the weight equations have different relationships?
3. How does changing units affect weight measurement?
4. What if additional constraints existed?
5. How to interpret the negative weights?

**Tip**: Always check plausible values before finalizing equations.

Learn how to solve a system of linear equations to determine the weights of large and small boxes used for fruit deliveries. This problem involves setting up and solving equations based on given delivery weights and box quantities. Suitable for high school students and those studying algebra.

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