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We are given a table with \(x\) and \(y\) values. To find the linear equation that relates \(x\) and \(y\), we will first find the slope \(m\), and then determine the equation in the form \(y = mx + b\), where \(b\) is the y-intercept.

### Step 1: Calculate the slope \(m\)
The slope formula is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can use any two points from the table. Let’s use the points \((4, -2)\) and \((5, -15)\).

\[
m = \frac{-15 - (-2)}{5 - 4} = \frac{-15 + 2}{1} = \frac{-13}{1} = -13
\]

So, the slope is \(m = -13\).

### Step 2: Find the y-intercept \(b\)
Now, use the equation \(y = mx + b\). We know \(m = -13\), and we can substitute one of the points from the table (for example, \((4, -2)\)) into the equation to find \(b\).

Substitute \(x = 4\), \(y = -2\), and \(m = -13\) into the equation:
\[
-2 = -13(4) + b
\]
\[
-2 = -52 + b
\]
\[
b = 50
\]

### Step 3: Write the final equation
Now that we have the slope \(m = -13\) and the y-intercept \(b = 50\), the equation is:
\[
y = -13x + 50
\]

This is the linear equation that describes the relationship between \(x\) and \(y\) for the table.

Would you like further details or have any other questions?

Here are 5 related questions to expand on this:
1. How would you verify the equation using the other points from the table?
2. What is the interpretation of the slope \(m = -13\) in this context?
3. How do we find the y-intercept if the values in the table don't include \(x = 0\)?
4. How can you graph the linear equation and what would it look like?
5. What if the table had more points—would this change the linear equation?

**Tip:** When working with a table of values, always check that the change in \(y\) values corresponds linearly with the change in \(x\) values before assuming the relationship is linear.

Write the linear equation that gives the rule for this table with values x = {4, 5, 6, 7} and y = {-2, -15, -28, -41}.

Learn how to find the linear equation for the given x and y values using the slope formula and the slope-intercept form. Solve for the slope and y-intercept step-by-step. This problem is suitable for students in Grades 7-10.

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