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To find the linear equation that represents the relationship between \( x \) and \( y \) in the given table, we can follow these steps:

### Table Analysis
The table provided is:

\[
\begin{array}{c|c}
x & y \\
\hline
4 & 9 \\
5 & 10 \\
6 & 11 \\
7 & 12 \\
\end{array}
\]

### Step 1: Find the slope (m)
The formula for the slope \( m \) of a linear equation \( y = mx + b \) is given by:

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the first two points \( (4, 9) \) and \( (5, 10) \):

\[
m = \frac{10 - 9}{5 - 4} = \frac{1}{1} = 1
\]

### Step 2: Find the y-intercept (b)
Now that we have the slope, we can use one of the points to find the y-intercept \( b \). Using the point \( (4, 9) \):

\[
y = mx + b
\]

Substitute \( m = 1 \), \( x = 4 \), and \( y = 9 \):

\[
9 = 1(4) + b
\]

\[
b = 9 - 4 = 5
\]

### Step 3: Write the equation
Now that we have \( m = 1 \) and \( b = 5 \), the equation of the line is:

\[
y = x + 5
\]

This is the linear equation that represents the data in the table.

Would you like more details, or do you have any questions about this solution?

Here are 5 related questions you might explore:
1. What happens to the equation if the slope was negative instead?
2. How would the equation change if the y-intercept were zero?
3. How do you determine if a set of points forms a linear relationship?
4. What are the implications of having a slope of zero?
5. Can a linear equation ever have a variable exponent greater than 1?

**Tip:** To verify your linear equation, you can substitute each \( x \) value from the table into the equation and check if you get the corresponding \( y \) value.

Write the linear equation that gives the rule for this table:

x | y
---|---
4 | 9
5 | 10
6 | 11
7 | 12

Write your answer as an equation with y first, followed by an equals sign.

Learn how to find the linear equation from a table of values, specifically focusing on the slope-intercept form. This problem involves key algebraic concepts such as calculating slope and determining the y-intercept, making it suitable for students in Grades 6-8.

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