To find the equation of the linear function that passes through the points (6, –7) and (9, –13/2), we first need to calculate the slope \( m \) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (6, -7)\) and \((x_2, y_2) = (9, -\frac{13}{2})\).
Substituting these coordinates into the formula:
\[ m = \frac{-\frac{13}{2} - (-7)}{9 - 6} \]
We can simplify this step by step. First, convert -7 to a fraction:
\[ -7 = -\frac{14}{2} \]
Now substituting back:
\[ m = \frac{-\frac{13}{2} + \frac{14}{2}}{3} = \frac{\frac{1}{2}}{3} = \frac{1}{6} \]
Thus, the slope \( m \) is \( \frac{1}{6} \).
Next, we use the point-slope form of the equation:
\[ y - y_1 = m(x - x_1) \]
Using the point (6, -7):
\[ y - (-7) = \frac{1}{6}(x - 6) \]
This simplifies to:
\[ y + 7 = \frac{1}{6}x - 1 \]
Now, subtract 7 from both sides:
\[ y = \frac{1}{6}x - 1 - 7 \]
This gives:
\[ y = \frac{1}{6}x - 8 \]
Thus, the equation of the linear function in the form \( y = mx + b \) is:
\[ \boxed{y = \frac{1}{6}x - 8} \]
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