This problem asks you to match each equation to its corresponding graph. Each equation given is a linear function of the form $y = mx$, where $m$ is the slope of the line. Let's go through the process of identifying the correct matches by observing the slopes from both the equations and the graphs.

Step 1: Equation Slopes

Below are the equations and their slopes:

• $a. \quad y = 2x \quad (m = 2)$
• $b. \quad y = \frac{4}{5}x \quad (m = \frac{4}{5})$
• $c. \quad y = \frac{1}{4}x \quad (m = \frac{1}{4})$
• $d. \quad y = \frac{2}{3}x \quad (m = \frac{2}{3})$
• $e. \quad y = \frac{3}{4}x \quad (m = \frac{3}{4})$
• $f. \quad y = \frac{3}{2}x \quad (m = \frac{3}{2})$

Step 2: Identify Graph Slopes

Now, we need to determine the rise over run (slope) from each graph.

1. Graph 1: Rise = 5, Run = 4 → Slope $m = \frac{5}{4}$. (This matches none of the given equations perfectly since no equation has $\frac{5}{4}x$).
2. Graph 2: Rise = 2, Run = 3 → Slope $m = \frac{2}{3}$. (Matches d. $y = \frac{2}{3}x$).
3. Graph 3: Rise = 3, Run = 2 → Slope $m = \frac{3}{2}$. (Matches f. $y = \frac{3}{2}x$).
4. Graph 4: Rise = 1, Run = 4 → Slope $m = \frac{1}{4}$. (Matches c. $y = \frac{1}{4}x$).
5. Graph 5: Rise = 3, Run = 4 → Slope $m = \frac{3}{4}$. (Matches e. $y = \frac{3}{4}x$).
6. Graph 6: Rise = 4, Run = 5 → Slope $m = \frac{4}{5}$. (Matches b. $y = \frac{4}{5}x$).

Step 3: Matching Results

Based on the above analysis:

• Graph 2 → $d. \quad y = \frac{2}{3}x$
• Graph 3 → $f. \quad y = \frac{3}{2}x$
• Graph 4 → $c. \quad y = \frac{1}{4}x$
• Graph 5 → $e. \quad y = \frac{3}{4}x$
• Graph 6 → $b. \quad y = \frac{4}{5}x$

Note: Graph 1 has a slope of $\frac{5}{4}$, which does not directly match any equation. However, it seems the closest matching equation might be $a. \quad y = 2x$, though the slope is not exactly aligned.

Final Matches:

• Graph 1: $a. \quad y = 2x$ (approximate match)
• Graph 2: $d. \quad y = \frac{2}{3}x$
• Graph 3: $f. \quad y = \frac{3}{2}x$
• Graph 4: $c. \quad y = \frac{1}{4}x$
• Graph 5: $e. \quad y = \frac{3}{4}x$
• Graph 6: $b. \quad y = \frac{4}{5}x$

Do you need any further clarification or help?
Here are five follow-up questions you might consider:

1. How do you calculate the slope between two points on a graph?
2. What would the equation look like if the slope were negative?
3. How can we determine the y-intercept of a line from a graph?
4. Why does a greater slope result in a steeper line?
5. What happens to the graph if we multiply the slope by a negative number?

Tip: To quickly estimate the slope from a graph, pick two points on the line, and use $m = \frac{\text{rise}}{\text{run}}$.