If x \(\begin{bmatrix}2\\3\end{bmatrix}\)+y \(\begin{bmatrix}-1\\1\end{bmatrix}\)=\(\begin{bmatrix}10\\5\end{bmatrix}\),find values of x and y.
If x \(\begin{bmatrix}2\\3\end{bmatrix}\)+y \(\begin{bmatrix}-1\\1\end{bmatrix}\)=\(\begin{bmatrix}10\\5\end{bmatrix}\),find values of x and y.
Solution and Explanation
x \(\begin{bmatrix}2\\3\end{bmatrix}\)+y \(\begin{bmatrix}-1\\1\end{bmatrix}\)=\(\begin{bmatrix}10\\5\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}2x\\3x\end{bmatrix}+\begin{bmatrix}-y\\y\end{bmatrix}\)=\(\begin{bmatrix}10\\5\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}2x-y\\3x+y\end{bmatrix}\)=\(\begin{bmatrix}10\\5\end{bmatrix}\)
Comparing the corresponding elements of these two matrices, we get:
2x − y = 10 and 3x + y = 5
Adding these two equations, we have:
5x = 15
\(\Rightarrow\) x = 3
Now, 3x + y = 5
\(\Rightarrow\) y = 5 − 3x
\(\Rightarrow\) y = 5 − 9 = −4
∴x = 3 and y = −4
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