Question

If $$\displaystyle x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$$Find values of $$x$$ and $$y$$.

Solution

$$\displaystyle x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$$$$\displaystyle \Rightarrow \left[ \begin{matrix} 2x \\ 3x \end{matrix} \right] +\left[ \begin{matrix} -y \\ y \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$$$$\displaystyle \Rightarrow \left[ \begin{matrix} 2x-y \\ 3x+y \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$$Comparing the corresponding elements of these two matrices, we get  $$\displaystyle 2x-y=10$$ and $$\displaystyle 3x+y=5$$Solving these we get,$$\displaystyle x=3\quad$$ and  $$\quad y=-4$$MathematicsNCERTStandard XII

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