To find the equation of the line that passes through the points \((-9, 7)\) and \((-6, -3)\), we can use the point-slope form of the equation of a line. First, we need to find the slope \( m \) of the line.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we can let:
- \((x_1, y_1) = (-9, 7)\)
- \((x_2, y_2) = (-6, -3)\)
Substituting these values into the slope formula:
\[ m = \frac{-3 - 7}{-6 - (-9)} = \frac{-10}{-6 + 9} = \frac{-10}{3} \]
Now that we have the slope \( m = -\frac{10}{3} \), we can use the point-slope form of the equation of a line which is:
\[ y - y_1 = m(x - x_1) \]
Using the point \((-9, 7)\):
\[ y - 7 = -\frac{10}{3}(x + 9) \]
Next, we simplify the equation:
\[ y - 7 = -\frac{10}{3}x - \frac{10}{3} \cdot 9 \] \[ y - 7 = -\frac{10}{3}x - 30 \]
Now, add 7 to both sides:
\[ y = -\frac{10}{3}x - 30 + 7 \] \[ y = -\frac{10}{3}x - 23 \]
Thus, the equation of the line in slope-intercept form is:
\[ \boxed{y = -\frac{10}{3}x - 23} \]
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