https://file.aiquickdraw.com/mathbot/file/1726164686124p9dmyatb.jpg

The table in the image shows pairs of \( x \) and \( y \) values:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & -6 \\
-3 & -3 \\
-2 & 0 \\
-1 & 3 \\
\hline
\end{array}
\]

### To determine if the relationship is linear:
A relationship is **linear** if the rate of change of \( y \) with respect to \( x \) (the slope) is constant.

1. Compute the change in \( y \) for each unit change in \( x \):
   - From \( x = -4 \) to \( x = -3 \): \( y \) changes from \( -6 \) to \( -3 \) → \( \Delta y = -3 - (-6) = 3 \).
   - From \( x = -3 \) to \( x = -2 \): \( y \) changes from \( -3 \) to \( 0 \) → \( \Delta y = 0 - (-3) = 3 \).
   - From \( x = -2 \) to \( x = -1 \): \( y \) changes from \( 0 \) to \( 3 \) → \( \Delta y = 3 - 0 = 3 \).

Since the change in \( y \) is consistent (always 3) for each unit change in \( x \), the relationship is **linear**.

### Now for the slope (rate of change):
The slope \( m \) is calculated as the ratio of the change in \( y \) to the change in \( x \):

\[
m = \frac{\Delta y}{\Delta x} = \frac{3}{1} = 3
\]

Therefore, the equation of the line can be written as \( y = 3x + b \), where \( b \) is the y-intercept. Using one of the points, say \( (-4, -6) \), to find \( b \):

\[
-6 = 3(-4) + b \implies -6 = -12 + b \implies b = 6
\]

So, the equation is \( y = 3x + 6 \).

Let me know if you want any further details! Here are some related questions you might find helpful:

1. How can you verify if other relationships are linear or non-linear?
2. How do you graph a linear equation like \( y = 3x + 6 \)?
3. How do you find the slope if the changes in \( x \) are not constant?
4. What happens if the slope between points changes?
5. How can you convert a non-linear equation to a linear form?

**Tip:** Always check for a constant rate of change to determine if a relation is linear!

Is the relationship between x and y linear or non-linear based on the table provided?

Analyze the relationship between x and y in a given table to determine if it is linear. The table includes values for x (-4, -3, -2, -1) and y (-6, -3, 0, 3), with a consistent slope of 3, indicating a linear relationship. Suitable for Grades 7-10.

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