Let's analyze the problem presented in the image. The linear equation given is:

$y = -\frac{3}{4}x + 2$

Two people, Aisha and Carolina, are graphing this equation using slightly different forms. Aisha uses the equation as it is ($y = -\frac{3}{4}x + 2$), while Carolina uses:

$y = \frac{-3}{4}x + 2$

This indicates that both equations are indeed identical, as the negative slope is represented consistently.

Part (a)

Explain how this leads them to use different steps to construct their graphs.

In both forms, the slope is $-\frac{3}{4}$ and the y-intercept is $2$. This means:

• The y-intercept, $b$, is $2$, so both Aisha and Carolina start at point $(0, 2)$ on the y-axis.
• The slope is $-\frac{3}{4}$, which means "down 3 units and right 4 units."

Aisha and Carolina might interpret these directions differently when plotting the next points, especially regarding the negative sign in the slope.

Part (b)

Will the two graphs look the same? Explain.

Yes, the graphs will look the same because both Aisha and Carolina are graphing the same linear equation, and hence the same slope and y-intercept apply. Even if the approaches seem different, the resulting line will be identical since the equation itself hasn't changed.

Choosing the Correct Answer

Let's evaluate each option:

• Option A: This suggests Aisha moves left 3 units and up 4 units while Carolina goes right 3 units and down 4 units. This does not correctly reflect the slope, which is $-\frac{3}{4}$ (meaning down 3 and right 4).

• Option B: States that Aisha identifies the y-intercept as $-3$ and Carolina as $3$. This is incorrect; the y-intercept for both should be $2$.

• Option C: States that Aisha identifies the y-intercept as $4$ and Carolina as $-4$. This is incorrect, as the y-intercept is clearly $2$.

• Option D: States that Aisha moves down $3$ units and right $4$ units while Carolina goes up $3$ units and left $4$. This could be a possible misinterpretation, but in reality, both should use the slope of $-\frac{3}{4}$, which always means "down $3$ and right $4$." Thus, this option seems to describe a scenario where they incorrectly interpret the direction.

Correct Answer: The correct interpretation is that both should end up plotting the same line, as the slope $-\frac{3}{4}$ and the y-intercept of $2$ do not change between the two forms.

The answer is not explicitly correct among the given options, but it seems likely that the question was testing their interpretation skills. Let me know if you need more detailed steps on any part.

Would you like to dive deeper into how to interpret slopes or need more examples? Here are some follow-up questions to expand on this topic:

1. How can you determine the slope from a graph without using the equation directly?
2. What are the steps to graph a line given in slope-intercept form?
3. How can you convert an equation from slope-intercept form to standard form?
4. What does a positive slope versus a negative slope indicate about a line?
5. How does changing the y-intercept affect the position of a line on a graph?

Tip: When interpreting slopes, always remember that a negative slope means the line decreases from left to right, while a positive slope means it increases from left to right.