Let's go through the questions in the image you uploaded.

Question 5:

Which of the following is represented by the slope of a linear equation?

The options are:

• A: The height of a rollercoaster for the duration of the ride
• B: The cost for one rose
• C: The weight of a dog
• D: The number of hours elapsed per day

Answer: The slope of a linear equation typically represents the rate of change between two variables. In this context, the most suitable option is D: The number of hours elapsed per day because it is something that occurs at a constant rate over time (i.e., time progresses uniformly). Other options like cost or weight might depend on multiple factors, and they aren't always modeled by a constant rate.

Question 6:

What is the slope of a line that passes through the points (-8, 14) and (4, -4)?

To find the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the formula:

$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Plugging in the points: $\text{slope} = \frac{-4 - 14}{4 - (-8)} = \frac{-18}{12} = -\frac{3}{2}$

Answer: The slope is -3/2, which corresponds to Option A.

Question 7:

A line with a slope of -3/4 passes through the points (-8, 2) and (4, a). What is the value of $a$?

We know the slope formula:

$\text{slope} = \frac{a - 2}{4 - (-8)} = \frac{a - 2}{12}$

We are told that the slope is $-\frac{3}{4}$, so we set the equation:

$\frac{a - 2}{12} = -\frac{3}{4}$

Multiplying both sides by 12:

$a - 2 = -9$

Solving for $a$:

$a = -9 + 2 = -7$

Answer: The value of $a$ is -7, corresponding to Option B.

Question 8:

For this question, you need to determine the slope and answer related questions regarding the similar triangles. Here is how to approach them:

Part A:

What is the slope of the solid line? Show your work.

To find the slope of the solid line, look at the points where the line crosses the grid. For example, the points $M(-6, 4)$ and $S(4, -4)$ seem to be on the line.

Using the slope formula again:

$\text{slope} = \frac{-4 - 4}{4 - (-6)} = \frac{-8}{10} = -\frac{4}{5}$

Answer: The slope of the solid line is $-\frac{4}{5}$.

Part B:

Does the slope of the line change between points $MN$ and points $RS$ for the two similar triangles? Explain your answer.

Since the triangles are similar, their corresponding sides are proportional, and the slope remains constant along the entire line. The slope of the line between $MN$ and $RS$ is the same because the line is straight, and similar triangles do not change the overall direction of the line.

Answer: No, the slope does not change. The slope remains $-\frac{4}{5}$ between points $MN$ and $RS$, as the triangles are similar and lie on the same straight line.

Would you like to go into more detail on any of these problems?

Here are 5 related questions you might want to consider:

1. How do similar triangles ensure that the slope of a line remains constant?
2. What is the difference between a positive and negative slope, and how does it affect the graph of a line?
3. How do you determine whether a relationship between two variables can be modeled by a linear equation?
4. If the points in question 6 were reversed, would the slope be different?
5. Why is the concept of slope important in understanding real-world situations, such as the number of hours elapsed per day?

Tip: Always check the signs of the coordinates when calculating slope to avoid errors in determining the rise and run.