Question

Find the values of x and y if (−3, −2) is the midpoint of the line connecting the points (−2, 1) and (x, y).

Answer 1

To find the values of \( x \) and \( y \), we can use the formula for the midpoint of a line segment. The midpoint \((M_x, M_y)\) of a line segment connecting the points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the following formulas:

\[
M_x = \frac{x_1 + x_2}{2}
\]
\[
M_y = \frac{y_1 + y_2}{2}
\]

In this case, we have:
- Point 1 \((x_1, y_1) = (-2, 1)\)
- Point 2 \((x_2, y_2) = (x, y)\)
- Midpoint \((M_x, M_y) = (-3, -2)\)

Using the midpoint formulas, we can set up the following equations:

1. For the x-coordinates:

\[
-3 = \frac{-2 + x}{2}
\]

Multiplying both sides by 2 to eliminate the fraction:

\[
-6 = -2 + x
\]

Now, adding 2 to both sides:

\[
-6 + 2 = x
\]
\[
x = -4
\]

2. For the y-coordinates:

\[
-2 = \frac{1 + y}{2}
\]

Again, multiplying both sides by 2 to eliminate the fraction:

\[
-4 = 1 + y
\]

Now, subtracting 1 from both sides:

\[
-4 - 1 = y
\]
\[
y = -5
\]

Thus, the values of \( x \) and \( y \) are:

\[
x = -4, \quad y = -5
\]

To summarize, the final result is:
\[
\boxed{(-4, -5)}
\]