To determine the equation of the linear function represented by the points in the table, we first need to calculate the slope (m) using the two given points: (18, 10) and (6, -3).
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Assign values from the points:
- \((x_1, y_1) = (18, 10)\)
- \((x_2, y_2) = (6, -3)\)
Now plug these values into the slope formula:
\[ m = \frac{-3 - 10}{6 - 18} = \frac{-13}{-12} = \frac{13}{12} \]
Now that we have the slope \(m = \frac{13}{12}\), we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using the point \((18, 10)\):
\[ y - 10 = \frac{13}{12}(x - 18) \]
Now distribute \( \frac{13}{12} \):
\[ y - 10 = \frac{13}{12}x - \frac{13}{12} \times 18 \] \[ y - 10 = \frac{13}{12}x - \frac{234}{12} \] \[ y - 10 = \frac{13}{12}x - \frac{39}{2} \]
Now, add 10 to both sides:
\[ y = \frac{13}{12}x - \frac{39}{2} + 10 \]
Convert 10 to a fraction with a denominator of 2:
\[ 10 = \frac{20}{2} \]
So:
\[ y = \frac{13}{12}x - \frac{39}{2} + \frac{20}{2} \] \[ y = \frac{13}{12}x - \frac{19}{2} \]
Thus, the equation of the linear function is:
\[ y = \frac{13}{12}x - \frac{19}{2} \]
This matches with the option:
y = (13/12)x - (-19/2)
Hence, the correct response is:
y = (13/12)x - (-19/2)