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Which equation represents the linear relationship between x and y

To determine which equation represents the linear relationship between \( x \) and \( y \), we can analyze the table of values shown in the image:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
y & 6 & 9 & 12 & 15 \\
\hline
\end{array}
\]

To find the equation that represents the relationship between \( x \) and \( y \), we need to observe how \( y \) changes as \( x \) increases. 

1. Calculate the slope (\( m \)) of the line using any two points. Let's use the points \( (1, 6) \) and \( (2, 9) \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 6}{2 - 1} = \frac{3}{1} = 3
\]

2. Now that we have the slope \( m = 3 \), we can use the point-slope form of the equation of a line, \( y = mx + b \), to find the y-intercept \( b \).

Using the point \( (1, 6) \):

\[
6 = 3(1) + b \implies b = 6 - 3 = 3
\]

So the equation of the line is:

\[
y = 3x + 3
\]

Looking at the options in the image, the correct equation that represents the linear relationship between \( x \) and \( y \) is:

\[
\boxed{y = 3x + 3}
\]

Let me know if you want further details or explanations! Here are some related questions:

1. How do you determine the slope from a table of values?
2. What does the y-intercept represent in a linear equation?
3. How can you verify that a linear equation correctly represents the data in a table?
4. What is the general form of a linear equation?
5. How can you interpret the meaning of slope in real-life scenarios?

**Tip:** When working with linear relationships, always check if the difference in \( y \) values divided by the difference in \( x \) values remains constant—this confirms that the relationship is indeed linear.

Learn how to identify and solve linear relationships between variables x and y. This example shows how to determine the equation of a line from a table of values, using concepts like slope and y-intercept. Suitable for Grades 7-9.

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