We are given the vectors:
\[ \mathbf{m} = \begin{pmatrix} 4 \ -10 \end{pmatrix}, \quad \mathbf{w} = \begin{pmatrix} 2x - 2 \ y + 5 \end{pmatrix}, \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} -2 \ 1 \end{pmatrix}. \]
We need to find the values of \( x \) and \( y \) given the relationship:
\[ \mathbf{r} = \frac{1}{2} \mathbf{m} + \mathbf{w}. \]
First, we compute \( \frac{1}{2} \mathbf{m} \):
\[ \frac{1}{2} \mathbf{m} = \frac{1}{2} \begin{pmatrix} 4 \ -10 \end{pmatrix} = \begin{pmatrix} 2 \ -5 \end{pmatrix}. \]
Next, we set up the equation according to the relationship provided:
\[ \mathbf{r} = \frac{1}{2} \mathbf{m} + \mathbf{w} \implies \begin{pmatrix} -2 \ 1 \end{pmatrix} = \begin{pmatrix} 2 \ -5 \end{pmatrix} + \begin{pmatrix} 2x - 2 \ y + 5 \end{pmatrix}. \]
This gives us two equations:
- First component:
\[ -2 = 2 + (2x - 2). \]
- Second component:
\[ 1 = -5 + (y + 5). \]
Let's solve these equations one by one.
Solving the First Component:
\[ -2 = 2 + 2x - 2 \]
Simplifying this:
\[ -2 = 2x. \]
Dividing both sides by 2, we get:
\[ x = -1. \]
Solving the Second Component:
\[ 1 = -5 + y + 5 \]
This simplifies to:
\[ 1 = y. \]
Final Values:
Thus, we have:
\[ x = -1, \quad y = 1. \]
The values of \( x \) and \( y \) are:
\[ \boxed{-1} \text{ and } \boxed{1}. \]