1. Solving the system of equations

We are given the system of equations:

1. $x + y = 20$
2. $7x + 4y = 104$

Where:

• $x$ represents the number of boxes of type $C_1$ (28 kg each),
• $y$ represents the number of boxes of type $C_2$ (16 kg each).

To solve the system, let's first use substitution or elimination.

Step 1: Solve the first equation for $y$:

From the equation $x + y = 20$, we can express $y$ as: $y = 20 - x$

Step 2: Substitute this expression for $y$ into the second equation:

Substitute $y = 20 - x$ into $7x + 4y = 104$: $7x + 4(20 - x) = 104$ Simplify: $7x + 80 - 4x = 104$ $3x + 80 = 104$ Subtract 80 from both sides: $3x = 24$ Solve for $x$: $x = 8$

Step 3: Find $y$:

Now substitute $x = 8$ into the equation $y = 20 - x$: $y = 20 - 8 = 12$

So, the solution to the system is:

• $x = 8$ (8 boxes of type $C_1$),
• $y = 12$ (12 boxes of type $C_2$).

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2. The mass problem

The second part of the problem is to determine how many boxes of each category there are, given:

• $C_1$ boxes weigh 28 kg each,
• $C_2$ boxes weigh 16 kg each,
• There are a total of 20 boxes, and their combined weight is 416 kg.

Let:

• $x$ represent the number of $C_1$ boxes,
• $y$ represent the number of $C_2$ boxes.

We now have the following system of equations:

1. $x + y = 20$ (since there are 20 boxes),
2. $28x + 16y = 416$ (since the total mass is 416 kg).

Step 1: Solve the first equation for $y$:

From $x + y = 20$, we can express $y$ as: $y = 20 - x$

Step 2: Substitute this expression into the second equation:

Substitute $y = 20 - x$ into $28x + 16y = 416$: $28x + 16(20 - x) = 416$ Simplify: $28x + 320 - 16x = 416$ $12x + 320 = 416$ Subtract 320 from both sides: $12x = 96$ Solve for $x$: $x = 8$

Step 3: Find $y$:

Now substitute $x = 8$ into $y = 20 - x$: $y = 20 - 8 = 12$

Thus, the solution to the second problem is:

• 8 boxes of type $C_1$ (28 kg each),
• 12 boxes of type $C_2$ (16 kg each).

Final Answer:

• For the first system of equations: $x = 8$ and $y = 12$.
• For the second problem: 8 boxes of type $C_1$ and 12 boxes of type $C_2$.

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Would you like further details or explanations on any step? Here are a few follow-up questions:

1. How would the solution change if the total number of boxes was different?
2. What would happen if the weight of one of the box types was increased?
3. How could this be solved using matrix methods?
4. What if the equations were more complex with more unknowns?
5. Can this system be solved graphically?

Tip: When solving systems of linear equations, sometimes substitution is faster, while elimination works well when the coefficients align neatly.