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Let's go through each part of the worksheet step by step.

### Problem 2
In problem 2, you need to find the slope (rate of change) for the linear relationship represented in the table. The table shows:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
y & 5 & 5.5 & 6 & 6.5 \\
\hline
\end{array}
\]

**Steps:**
1. Choose two points from the table. For instance, you could use \((0, 5)\) and \((1, 5.5)\).
2. Apply the slope formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).

Using the points \((0, 5)\) and \((1, 5.5)\):
\[
\text{slope} = \frac{5.5 - 5}{1 - 0} = \frac{0.5}{1} = 0.5
\]

**Answer:** The slope is \(0.5\).

### Problem 3
In problem 3, the table shows:

\[
\begin{array}{|c|c|c|c|}
\hline
x & 3 & 8 & 14 \\
y & 15 & 15 & 15 \\
\hline
\end{array}
\]

The \(y\)-values are constant at \(15\), meaning there is no change in \(y\) as \(x\) changes. Thus, the slope is \(0\), indicating a horizontal line.

**Answer:** The slope is \(0\).

### Problem 4
A web designer charges a consultation fee and an hourly rate, represented by the table:

\[
\begin{array}{|c|c|}
\hline
\text{Hours} & \text{Total Cost} \\
2 & 205 \\
5 & 400 \\
8 & 595 \\
10 & 725 \\
\hline
\end{array}
\]

#### (a) Find the rate of change.
Choose two points, say \((2, 205)\) and \((5, 400)\), and apply the slope formula.

\[
\text{slope} = \frac{400 - 205}{5 - 2} = \frac{195}{3} = 65
\]

**Answer:** The rate of change (slope) is \(65\).

#### (b) Interpretation of the Rate of Change
The rate of change represents the hourly rate charged by the web designer, as each hour increases the total cost by \(65\).

### Problem 5
The task is to find the rate of change of shipping cost per pound based on different weights and costs provided.

Choose two points based on the information in the problem description, and apply the slope formula to determine the rate of change. This rate represents the cost per pound added for shipping.

### Problem 6
For the graph, Jude and Yin each chose two points to find the slope using their right triangles. 

#### (a) Using Jude's points
Find the coordinates of the points Jude chose, calculate the rise over run, and simplify.

#### (b) Using Yin's points
Similarly, find Yin's points, calculate the slope, and simplify.

#### (c) Relationship Between the Triangles
The triangles should be similar because they share the same slope, representing the same line.

#### (d) Slope Consistency
Since the slope is the same for any two points on the same line, this confirms a constant rate of change for a linear relationship.

---

Would you like detailed calculations for any of these problems or further explanations? 

### Related Questions
1. How can you identify a linear relationship from a table of values?
2. What does a slope of zero indicate about a line?
3. How can you interpret the slope in real-life scenarios?
4. Why are triangles on a line graph similar when finding slopes?
5. What happens to the slope if the line were not straight?

### Tip
Remember, the slope formula \(\frac{y_2 - y_1}{x_2 - x_1}\) only works for linear relationships where there is a constant rate of change.

This worksheet includes several problems on finding slopes and interpreting linear relationships. Specifically:

1. Problems 2 and 3 ask students to determine the slope (rate of change) from a table of values by selecting pairs of points and applying the slope formula.
2. Problem 4 requires students to interpret the rate of change for a web designer's fee structure.
3. Problem 5 involves finding the rate of change in shipping costs.
4. Problem 6 involves interpreting slope using a graph with triangles and comparing slopes derived from different points.

The user seeks step-by-step guidance on solving these problems.

This worksheet covers calculating slope from tables and graphs, interpreting rate of change in context, and understanding linear relationships. It includes problems on finding slope between points, analyzing hourly rates, and graph-based slope interpretation for students in Grades 7-9.

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