To determine which equation matches the given graph, let's analyze the line visually.

Step 1: Analyze the slope

The line in the image has a negative slope, as it is slanting downward from left to right. This tells us that the equation of the line must have a negative coefficient for $x$ if written in the form $y = mx + b$.

Step 2: Analyze the y-intercept

The line crosses the y-axis at a positive point, indicating that the y-intercept $b$ in the equation $y = mx + b$ is positive.

Step 3: Check the options

Let's check the given equations and identify which could represent this line:

• A. $y = -4x + 2$: This equation has a negative slope (-4) and a positive y-intercept (2), which matches our analysis. So, Option A could be correct.

• B. $3x + 2y = 6$: Rewriting this equation in slope-intercept form $y = mx + b$, we solve for $y$: $2y = -3x + 6 \quad \Rightarrow \quad y = -\frac{3}{2}x + 3$ The slope is negative $\left( -\frac{3}{2} \right)$, and the y-intercept is 3, so this could also match the graph. Option B could be correct.

• C. $y = 5$: This represents a horizontal line with slope 0, which does not match the graph. Option C is not correct.

• D. $2x - y = 7$: Rewriting this in slope-intercept form: $-y = -2x + 7 \quad \Rightarrow \quad y = 2x - 7$ This equation has a positive slope, so it does not match the graph. Option D is not correct.

Conclusion:

The equations that match the graph are A and B.

Would you like further details or explanations? Here are 5 follow-up questions:

1. How do we determine the slope from a graph?
2. Why does a negative slope indicate a line slanting downward?
3. What are the general forms of linear equations, and how can we convert between them?
4. How do y-intercepts help in identifying equations?
5. Can you explain how to solve for $y$ from a standard-form linear equation?

Tip: Always check both the slope and y-intercept when matching an equation to a graph.