The problem asks which graph represents the equation $-8x + 5y = 32$.

To determine the correct graph, let's first rewrite the equation in slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Step 1: Solve for $y$

Starting with: $-8x + 5y = 32$ Add $8x$ to both sides: $5y = 8x + 32$ Now, divide by 5: $y = \frac{8}{5}x + \frac{32}{5}$

Step 2: Interpret the slope and intercept

• The slope is $\frac{8}{5}$, which means the line rises by 8 units for every 5 units it moves to the right.
• The y-intercept is $\frac{32}{5}$, or 6.4.

Step 3: Analyze the graphs

• The correct graph should have a y-intercept at approximately 6.4 and a slope of $\frac{8}{5}$.

Looking at the graphs:

• Graph A: The slope is negative and does not match the calculated slope.
• Graph B: This graph has a steep slope and a large y-intercept but does not match the positive slope of $\frac{8}{5}$.
• Graph C: This graph has a positive slope and a y-intercept near 6.4. Let's check if the slope matches $\frac{8}{5}$, and it seems correct.
• Graph D: The slope is steeper than $\frac{8}{5}$.

Conclusion:

Graph C represents the equation $-8x + 5y = 32$ because it has the correct slope and y-intercept.

Would you like further explanation or have any questions?

Here are 5 related questions to deepen understanding:

1. How do you find the slope of a line from two points on a graph?
2. What is the significance of the y-intercept in a linear equation?
3. How does changing the slope affect the steepness of a line?
4. What happens to the graph of a line if you change the y-intercept?
5. How can you verify a graph represents an equation by substituting points?

Tip: Always convert equations to slope-intercept form $y = mx + b$ when analyzing graphs!