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To determine the linear equation that fits the given table of values, we can analyze the changes in \( x \) and \( y \):

The table shows the following pairs:

\[
\begin{array}{c|c}
x & y \\
\hline
5 & 7 \\
6 & 8 \\
7 & 9 \\
8 & 10 \\
\end{array}
\]

We observe that as \( x \) increases by 1, \( y \) also increases by 1. This suggests that the slope \( m \) of the linear equation \( y = mx + b \) is 1.

Now, let's find the y-intercept \( b \). Using one of the points, such as \( (5, 7) \), we substitute the values into the equation:

\[
7 = 1(5) + b
\]

\[
7 = 5 + b
\]

\[
b = 7 - 5 = 2
\]

Thus, the linear equation is:

\[
y = x + 2
\]

If you have any questions or would like more details, feel free to ask!

Here are 5 related questions to expand on this concept:

1. How can you determine if a set of points forms a linear relationship?
2. What does the slope of a linear equation represent in real-world contexts?
3. How do you find the slope between two points on a coordinate plane?
4. What happens to the graph of a line when you change the y-intercept?
5. Can the linear equation \( y = x + 2 \) be expressed in a different form?

**Tip:** When working with linear equations, always check the change in \( y \) relative to the change in \( x \) to find the slope quickly.

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

Learn how to derive the linear equation from a given table of values. This problem demonstrates key concepts in algebra, including the slope-intercept form and calculating the y-intercept. Suitable for students in Grades 7-9.

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