To find the equation of the linear function represented by the two points \((18, 10)\) and \((6, -3)\), we first need to determine the slope \(m\) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where \((x_1, y_1) = (18, 10)\) and \((x_2, y_2) = (6, -3)\).
Substituting the values in:
\[ m = \frac{-3 - 10}{6 - 18} = \frac{-13}{-12} = \frac{13}{12} \]
Now that we have the slope \(m = \frac{13}{12}\), we can use one of the points to find the y-intercept \(b\). We'll use the point \((18, 10)\).
The equation of the line is given by:
\[ y = mx + b \]
Substituting the slope and the coordinates of the point into the equation:
\[ 10 = \frac{13}{12}(18) + b \]
Calculating \(\frac{13}{12} \times 18\):
\[ \frac{13 \times 18}{12} = \frac{234}{12} = 19.5 \]
Now inputting this back into the equation:
\[ 10 = 19.5 + b \]
Solving for \(b\):
\[ b = 10 - 19.5 = -9.5 = -\frac{19}{2} \]
Thus the equation of the line is:
\[ y = \frac{13}{12}x - \frac{19}{2} \]
Now, looking at the options given:
- \(y=\frac{12}{13}x - \frac{111}{13}\)
- \(y=\frac{13}{12}x - \frac{19}{2}\)
- \(y=-\frac{19}{2}\)
- \(y=\left(-\frac{13}{12}\right)x + \frac{7}{2}\)
The correct response is:
y=\frac{13}{12}x - \frac{19}{2}