https://file.aiquickdraw.com/mathbot/file/1725411699978d5pzuvls.jpg

Let's analyze the image step by step.

### Task A: Complete the Table
The equation provided is \(2x + y = 9\). The task is to find the values of \(y\) for different \(x\) values, and to find the coordinates \((x, y)\).

The table has the following columns:
- \(x\) values: -2, 0, 4, 8
- The corresponding \(y\) values need to be calculated.
- Finally, we list the \((x, y)\) pairs.

#### Step-by-Step Solution:
1. **For \(x = -2\):**
   \[
   2(-2) + y = 9 \\
   -4 + y = 9 \\
   y = 13
   \]
   Coordinate: \((-2, 13)\)

2. **For \(x = 0\):**
   \[
   2(0) + y = 9 \\
   y = 9
   \]
   Coordinate: \((0, 9)\)

3. **For \(x = 4\):**
   \[
   2(4) + y = 9 \\
   8 + y = 9 \\
   y = 1
   \]
   Coordinate: \((4, 1)\)

4. **For \(x = 8\):**
   \[
   2(8) + y = 9 \\
   16 + y = 9 \\
   y = -7
   \]
   Coordinate: \((8, -7)\)

So, the completed table is:

\[
\begin{array}{|c|c|c|}
\hline
x & y & (x, y) \\
\hline
-2 & 13 & (-2, 13) \\
0 & 9 & (0, 9) \\
4 & 1 & (4, 1) \\
8 & -7 & (8, -7) \\
\hline
\end{array}
\]

### Task B: Determine the Slope
The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slopes for each pair:

1. **For \((2, 4)\) and \((1, 6)\):**
   \[
   m = \frac{6 - 4}{1 - 2} = \frac{2}{-1} = -2
   \]

2. **For \((3, -1)\) and \((5, 3)\):**
   \[
   m = \frac{3 - (-1)}{5 - 3} = \frac{4}{2} = 2
   \]

3. **For \((-3, 2)\) and \((-3, -3)\):**
   \[
   m = \frac{-3 - 2}{-3 - (-3)} = \frac{-5}{0} = \text{undefined}
   \]
   (The slope is undefined because the line is vertical.)

4. **For \((-3, -1)\) and \((2, 2)\):**
   \[
   m = \frac{2 - (-1)}{2 - (-3)} = \frac{3}{5} = \frac{3}{5}
   \]

5. **For \((5, 7)\) and \((2, 2)\):**
   \[
   m = \frac{2 - 7}{2 - 5} = \frac{-5}{-3} = \frac{5}{3}
   \]

### Summary of Slopes:
1. -2
2. 2
3. Undefined
4. \( \frac{3}{5} \)
5. \( \frac{5}{3} \)

Would you like more details or have any other questions?

### Follow-up Questions:
1. How do you interpret the slope of a vertical line?
2. What happens to the slope if two points have the same \(y\)-coordinate?
3. Can you find the slope of a line given an equation in standard form?
4. How would the table in Task A change if the equation was \(2x + y = 0\)?
5. How can the slope formula be derived from the equation of a line?

### Tip:
Remember, the slope of a line gives the rate at which \(y\) changes with respect to \(x\). It’s a measure of the steepness and direction of the line.

Complete the table with the given equation and determine the slope of the line between given points.

Complete the table using the linear equation 2x + y = 9 and calculate slopes for given pairs of points. This activity covers key algebraic concepts and coordinate geometry. Suitable for students in Grades 7-9.

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