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each table below represents a linear equation. write the equation of the line then use it to complete each table

Let's solve this step by step by deriving the linear equation that connects \(x\) and \(y\) from the given values.

The table provides the following points:
- \((-4, 4)\)
- \((-1, 8)\)
- \((8, 20)\)
- \((17, ?)\)

### Step 1: Determine the slope \(m\)
The slope of a line is calculated by using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-4, 4)\) and \((-1, 8)\):
\[
m = \frac{8 - 4}{-1 - (-4)} = \frac{4}{3}
\]

### Step 2: Find the equation of the line
Now that we know the slope is \(\frac{4}{3}\), we use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((-1, 8)\), we substitute \(x_1 = -1\), \(y_1 = 8\), and \(m = \frac{4}{3}\):
\[
y - 8 = \frac{4}{3}(x + 1)
\]
Simplifying:
\[
y - 8 = \frac{4}{3}x + \frac{4}{3}
\]
\[
y = \frac{4}{3}x + 8 + \frac{4}{3}
\]
\[
y = \frac{4}{3}x + \frac{28}{3}
\]

### Step 3: Complete the table
We use the equation \(y = \frac{4}{3}x + \frac{28}{3}\) to find the missing \(y\)-values for \(x = 17\).

Substitute \(x = 17\):
\[
y = \frac{4}{3}(17) + \frac{28}{3} = \frac{68}{3} + \frac{28}{3} = \frac{96}{3} = 32
\]

Thus, the completed table is:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & 4 \\
-1 & 8 \\
8 & 20 \\
17 & 32 \\
\hline
\end{array}
\]

Would you like further details or clarifications?

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Here are 5 related questions to expand your understanding:
1. How do you calculate the slope when the points have different coordinates?
2. What is the point-slope form of a line, and how does it relate to the slope-intercept form?
3. How can we use the equation of a line to predict values of \(y\) for any given \(x\)?
4. Can a table always represent a linear relationship? What conditions must be met?
5. How do you verify that a set of points lies on the same line?

**Tip**: Always double-check your calculations, especially when converting between fractions and decimals!

Each table below represents a linear equation. Write the equation of the line then use it to complete each table.

Learn how to solve linear equations by finding the slope and using the point-slope form. This example uses a table with points (-4, 4), (-1, 8), and (8, 20) to derive the equation and complete missing values. Suitable for students in Grades 8-10.

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