To solve this problem graphically, let’s first set up the system of equations based on the information provided:

1. Define the variables:

• $x$: Number of small boxes shipped
• $y$: Number of large boxes shipped

2. Write the equations:

• Each small box weighs 50 pounds, and each large box weighs 75 pounds. The total weight is 1200 pounds: $50x + 75y = 1200$

• There were twice as many large boxes as small boxes: $y = 2x$

3. Solve the system graphically:

We will graph both equations on the same coordinate plane to find the intersection point, which represents the solution.

Rearrange the equations for graphing:

1. From the first equation: $y = \frac{1200 - 50x}{75}$

2. The second equation is already solved for $y$: $y = 2x$

Let me plot these equations. numpy as np import matplotlib.pyplot as plt

Define x values for the graph

x_values = np.linspace(0, 30, 300) # Choose a range for x

First equation: y = (1200 - 50x) / 75

y1_values = (1200 - 50 * x_values) / 75

Second equation: y = 2x

y2_values = 2 * x_values

Plot the equations

plt.figure(figsize=(8, 6)) plt.plot(x_values, y1_values, label=r"$50x + 75y = 1200$") plt.plot(x_values, y2_values, label=r"$y = 2x$")

Add labels and title

plt.title("Graphical Solution of the System of Equations") plt.xlabel("Number of Small Boxes (x)") plt.ylabel("Number of Large Boxes (y)")

Add grid and legend

plt.grid(True) plt.legend()

Limit the axes to focus on the solution

plt.xlim(0, 30) plt.ylim(0, 70)

plt.show()From the graph, the solution to the system of equations is the point of intersection. Based on the equations, you can see where the two lines meet.

Would you like me to calculate the exact values for $x$ and $y$, or do you have specific questions about interpreting this graph? Here are some related questions you might explore:

1. What are the coordinates of the intersection point?
2. How can you verify the solution algebraically?
3. What would happen if the weight of the small and large boxes changed?
4. How does the constraint $y = 2x$ affect the solution?
5. Could there be any real-life considerations affecting this shipment?

Tip: Graphing equations is an excellent way to visualize solutions, but always double-check calculations for precision.