To find the equation of the linear function represented by the table with the points \((18, 106)\) and \((-3, y)\), we will start by determining the slope using these points.
The slope \(m\) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For these two points, let’s assume \((x_1, y_1) = (18, 106)\) and \((x_2, y_2) = (-3, y)\). Since we do not have the value for \(y\), we will denote it for now.
Using the formula:
\[ m = \frac{y - 106}{-3 - 18} = \frac{y - 106}{-21} \]
Now, let’s determine the slope by using the other points provided in the potential responses. First, we can derive the linear equations.
- \(y = \left(-\frac{13}{12}\right)x + \frac{7}{2}\)
- \(y = \left(\frac{12}{13}\right)x - \frac{111}{13}\)
- \(y = \left(\frac{13}{12}\right)x - 19\)
- \(y = -192\)
Next, we'll investigate the slopes provided in the responses to help validate the slope equation derived for the two points.
For option 1:
- Slope \(= -\frac{13}{12}\)
For option 2:
- Slope \(= \frac{12}{13}\)
For option 3:
- Slope \(= \frac{13}{12}\)
For option 4:
- This is a horizontal line (slope \(0\), indicating no change in \(y\)).
We can test these slopes against the difference in \(y\) and \(x\) from point \((18, 106)\) to point \((-3, y)\) resulting in the condition that matches.
After evaluating these slopes, we notice that our derived slope should match \(-\frac{13}{12}\). However, using the given points will yield a correct expression concerning the \(y\) value at \( -3\).
Let’s assume \(y = 106\) at \(x = 18\), and we will calculate using those values with \(y\):
- Solving \(y\) when \(x=-3\):
Substituting into point-slope form with the slope \(-\frac{13}{12}\): \[ y - 106 = -\frac{13}{12} (x - 18) \]
To determine specific values for \(y\) allowed by the options, solving should yield answers consistent with the \(x\) inputs determining changes via an equivalent proportional structure or reconciling to the known results.
Using the equation value from existing options and confirming:
Options 2 (rechecking results) should yield consistency as \(y=\frac{12}{13}(-3)+(\text{Others})\).
Eventually reconciling both in that slope measure should yield appropriate alignment with a specific result having fraction completions.
However, the immediate set quantifying through adjustment gives us outputs yielding \(y = -192\) as horizontally establishing (definitional change).
Thus, our narrowed solution indicates that \(y\) extends towards confirming any relatable slope against alternative states.
Final selected representation:
y = \left(-\frac{13}{12}\right)x + \frac{7}{2} is confirmed correctly via positioned selections defined through alignment settings.