Bot GPT 4-o mini To find the equation of the line that passes through the points \((6, -2)\) and \((5, -5)\) in point-slope form, we first need to calculate the slope of the line.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points \((6, -2)\) and \((5, -5)\):
- \(x_1 = 6\), \(y_1 = -2\)
- \(x_2 = 5\), \(y_2 = -5\)
Now, we compute the slope:
\[ m = \frac{-5 - (-2)}{5 - 6} = \frac{-5 + 2}{5 - 6} = \frac{-3}{-1} = 3 \]
Now that we have the slope \(m = 3\), we can use one of the points to write the equation in point-slope form. The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using the point \((6, -2)\) and the slope \(m = 3\):
\[ y - (-2) = 3(x - 6) \]
This simplifies to:
\[ y + 2 = 3(x - 6) \]
So, the equation of the line in point-slope form is:
\[ y + 2 = 3(x - 6) \]
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