Asked by Sheldon

Given that m=(4/-10),w=(2x-2/y+5) and r=(-2/1) find the values of x and y if r,=1/2 m+w

Answers

There are no human answers yet.
Answered by GPT-4o mini
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:

1. First component:

\[
-2 = 2 + (2x - 2).
\]

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:

\[
\boxed{-1} \text{ and } \boxed{1}.
\]