To find the midpoint \( M \) of the line segment joining the points \( P = (-7, -5) \) and \( Q = (5, -1) \), we use the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( (x_1, y_1) = (-7, -5) \) and \( (x_2, y_2) = (5, -1) \).
Now, we can calculate the coordinates of the midpoint \( M \):
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Calculate the x-coordinate of \( M \): \[ \frac{x_1 + x_2}{2} = \frac{-7 + 5}{2} = \frac{-2}{2} = -1 \]
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Calculate the y-coordinate of \( M \): \[ \frac{y_1 + y_2}{2} = \frac{-5 + (-1)}{2} = \frac{-5 - 1}{2} = \frac{-6}{2} = -3 \]
Thus, the midpoint \( M \) is given by:
\[ M = (-1, -3) \]
In conclusion, the midpoint of the line segment joining the points \( P \) and \( Q \) is \( M = (-1, -3) \).