To solve this problem, we need to match each graph with its corresponding equation based on the slope.

Let's analyze each given graph and equation:

1. Equation Details:

   • $y = 2x$: The slope (rate of increase) is 2, meaning it goes up 2 units for every 1 unit to the right.
   • $y = \frac{3}{2}x$: The slope is $\frac{3}{2}$, meaning it goes up 3 units for every 2 units to the right.
   • $y = \frac{1}{4}x$: The slope is $\frac{1}{4}$, meaning it goes up 1 unit for every 4 units to the right.
   • $y = \frac{4}{3}x$: The slope is $\frac{4}{3}$, meaning it goes up 4 units for every 3 units to the right.

2. Matching each graph:

   • Graph A: The slope here is steep, suggesting it matches with $y = 2x$.
   • Graph B: The slope here is very shallow, which corresponds to $y = \frac{1}{4}x$.
   • Graph C: This slope is somewhat steep but not as much as in Graph A, so it matches with $y = \frac{3}{2}x$.
   • Graph D: This slope is steeper than B but not as steep as A or C, fitting $y = \frac{4}{3}x$.

3. Final Matching:

   • Graph A → $y = 2x$
   • Graph B → $y = \frac{1}{4}x$
   • Graph C → $y = \frac{3}{2}x$
   • Graph D → $y = \frac{4}{3}x$

Let me know if you need further clarification on any of these points.

Follow-up Questions

1. How is the slope of a line determined from a graph?
2. What does a positive slope indicate about the direction of a line?
3. How would the graphs look if the slope was negative?
4. How do changes in the slope value affect the steepness of the line?
5. What would happen to the line if a constant was added to the equation?

Tip

When matching equations to graphs, focus on the slope to quickly identify the correct match.