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To write the linear equation that represents the data in the table, we need to determine the relationship between \(x\) and \(y\). Here's the table provided:

| \(x\) | \(y\)  |
|:----:|:-----:|
|  5   |  -22  |
|  6   |  -34  |
|  7   |  -46  |
|  8   |  -58  |

### Step 1: Find the slope (rate of change)
The formula for the slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((5, -22)\) and \((6, -34)\):
\[
m = \frac{-34 - (-22)}{6 - 5} = \frac{-34 + 22}{1} = \frac{-12}{1} = -12
\]

So, the slope is \(-12\).

### Step 2: Find the y-intercept (\(b\))
The general form of a linear equation is:
\[
y = mx + b
\]
We can use one of the points, say \((5, -22)\), and the slope \(m = -12\) to solve for \(b\).

\[
-22 = (-12)(5) + b
\]
\[
-22 = -60 + b
\]
\[
b = -22 + 60 = 38
\]

### Step 3: Write the equation
Now that we have \(m = -12\) and \(b = 38\), the equation is:
\[
y = -12x + 38
\]

Thus, the linear equation that gives the rule for this table is:
\[
y = -12x + 38
\]

Would you like any further details or clarifications?

Here are five related questions to deepen your understanding:
1. How would the equation change if the slope were positive instead of negative?
2. How can you check if a set of points follows a linear pattern using a graph?
3. What does the slope of a line represent in real-world scenarios?
4. How can you determine the y-intercept just by looking at a graph?
5. What is the significance of the y-intercept in different types of linear equations?

**Tip**: When writing a linear equation, always verify both the slope and y-intercept by substituting a point from the table back into the equation.

Write the linear equation that gives the rule for this table: x = {5, 6, 7, 8} and y = {-22, -34, -46, -58}.

Learn how to write a linear equation from a table with values for x = {5, 6, 7, 8} and y = {-22, -34, -46, -58}. This problem involves finding the slope and y-intercept to create the linear equation y = -12x + 38, suitable for 8th-grade students.

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