The problem presented involves a paper company that needs to ship paper in small boxes and large boxes, and the weights of the boxes are given. Here's a breakdown of the key information:

• Each small box of paper weighs 50 pounds.
• Each large box of paper weighs 75 pounds.
• There were twice as many large boxes shipped as small boxes.
• The total weight of all the boxes shipped was 1,200 pounds.

We need to determine:

• The number of small boxes shipped ($x$).
• The number of large boxes shipped ($y$).

The problem can be modeled using a system of equations. Let's set up the equations:

1. The total weight equation:
   $50x + 75y = 1200$ This represents the total weight of all the boxes shipped, where $x$ is the number of small boxes and $y$ is the number of large boxes.

2. The relationship between the number of small and large boxes:
   $y = 2x$ This states that there are twice as many large boxes as small boxes.

Solution:

To solve this system graphically, we can substitute the second equation into the first:

Substitute $y = 2x$ into the first equation: $50x + 75(2x) = 1200$ Simplifying: $50x + 150x = 1200$ $200x = 1200$ $x = \frac{1200}{200} = 6$

Now, substitute $x = 6$ into the equation $y = 2x$: $y = 2(6) = 12$

Conclusion:

• The number of small boxes shipped ($x$) is 6.
• The number of large boxes shipped ($y$) is 12.

Would you like a detailed graph to illustrate this, or do you have any additional questions?

Here are five related questions to expand on this topic:

1. How would the system change if the weights of the boxes were different?
2. What if the number of large boxes was three times the number of small boxes?
3. How would you solve the system algebraically without graphing?
4. Can you set up a similar problem with different total weights and box quantities?
5. How does the method of substitution compare to solving by elimination?

Tip: When graphing linear equations, always isolate one variable to easily plot points and understand the slope of the line.