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To find the linear equation for the table, we start by analyzing the relationship between \( x \) and \( y \) values given in the table. 

The table shows:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 6 \\
4 & 13 \\
5 & 20 \\
6 & 27 \\
\hline
\end{array}
\]

### Step 1: Calculate the Slope
The slope \( m \) of a linear function \( y = mx + b \) can be calculated using two points from the table. For example, we’ll use the points \( (3, 6) \) and \( (4, 13) \).

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - 6}{4 - 3} = \frac{7}{1} = 7
\]

So, the slope \( m = 7 \).

### Step 2: Find the y-intercept
Now that we know the slope, we can use one of the points to find the y-intercept \( b \). Let's use the point \( (3, 6) \):

\[
y = mx + b \implies 6 = 7 \cdot 3 + b
\]
\[
6 = 21 + b
\]
\[
b = 6 - 21 = -15
\]

### Step 3: Write the Equation
The equation of the line is:
\[
y = 7x - 15
\]

This is the linear equation that describes the rule for this table.

Would you like further details, or do you have any questions?

Here are five related questions to deepen understanding:
1. How do you calculate the slope if you’re given two different points?
2. How can you confirm if a table of values follows a linear pattern?
3. What happens to the equation if the slope is zero?
4. How can you use this equation to predict \( y \) values for other \( x \) values?
5. Why is it important to find the y-intercept in a linear equation?

**Tip:** When working with tables, check that the differences between \( y \)-values remain consistent with each increase in \( x \)-value to confirm linearity.

Write the linear equation that gives the rule for this table. Write your answer as an equation with y first, followed by an equals sign.

Learn how to determine the linear equation that represents a given table of values. This problem involves calculating the slope and y-intercept to write the equation in the form y = mx + b. Suitable for students in Grades 7-9.

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