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
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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:
\[
\boxed{-1} \text{ and } \boxed{1}.
\]
\[ \mathbf{m} = \begin{pmatrix} 4 \\ -10 \end{pmatrix}, \quad \mathbf{w} = \begin{pmatrix} 2x - 2 \\ y + 5 \end{pmatrix}, \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} -2 \\ 1 \end{pmatrix}. \]
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:
\[
\boxed{-1} \text{ and } \boxed{1}.
\]
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