To find the rule for the linear equation based on the given values in the table, we'll first identify the relationship between \(x\) and \(y\).
From the table, we have the following pairs:
- When \(x = 3\), \(y = -36\)
- When \(x = 4\), \(y = -48\)
- When \(x = 5\), \(y = -60\)
- When \(x = 6\), \(y = -72\)
To determine the pattern, let's look at the change in \(y\) as \(x\) increases.
- From \(x = 3\) to \(x = 4\), \(y\) changes from \(-36\) to \(-48\) (change of \(-12\)).
- From \(x = 4\) to \(x = 5\), \(y\) changes from \(-48\) to \(-60\) (change of \(-12\)).
- From \(x = 5\) to \(x = 6\), \(y\) changes from \(-60\) to \(-72\) (change of \(-12\)).
Here, we can see that as \(x\) increases by \(1\), \(y\) decreases by \(12\). This indicates a slope of \(-12\).
We can express the linear equation in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
So, starting with:
\[ y = -12x + b \]
Now, we can find \(b\) (the y-intercept) by substituting one of the values from the table. Let's use \(x = 3\) and \(y = -36\):
\[ -36 = -12(3) + b \] \[ -36 = -36 + b \] \[ b = 0 \]
Now, we can write the complete equation:
\[ y = -12x + 0 \]
So, the final rule based on the table is:
\[ y = -12x \]
Thus, filling in the missing values, we have:
y = -12x + 0
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