X तथा Y ज्ञात कीजिए यदि 2X + 3Y = $\left[\begin{array}{ll} 2 & 3 \\ 4 & 0 \end{array}\right]$तथा 3X + 2Y =$\left[\begin{array}{rr} 2 & -2 \\ -1 & 5 \end{array}\right]$
Exercise-3.2-7(2)
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दिया है,
2X + 3Y = $ \left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$ ...(i)
तथा 3X + 2Y = $\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right]$...(ii)
समी (i) को 2 से तथा समी (ii) को 3 से गुणा करके परस्पर घटाने पर,
2(2X + 3Y) - 3(3X + 2Y) = 2$\left[\begin{array}{ll} 2 & 3 \\ 4 & 0 \end{array}\right]$ - 3$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 5 \end{array}\right]$
$ \Rightarrow $ 4X + 6Y - 9X - 6Y = $\left[\begin{array}{ll} 4 & 6 \\ 8 & 0 \end{array}\right]$ - $\left[\begin{array}{rr} 6 & -6 \\ -3 & 15 \end{array}\right]$
$ \Rightarrow $ - 5X = $\left[\begin{array}{ll} 4-6 & 6+6 \\ 8+3 & 0-15 \end{array}\right]$= $\left[\begin{array}{rr} -2 & 12 \\ 11 & -15 \end{array}\right] $
$ \Rightarrow $ X = - $ \frac{1}{5}$ $\left[\begin{array}{rr} -2 & 12 \\ 11 & -15 \end{array}\right]$ = $ \left[\begin{array}{rr} 2 / 5 & -12 / 5 \\ -11 / 5 & 3 \end{array}\right] $
तब समी (i) से, 3Y = $ \left[\begin{array}{ll} 2 & 3 \\ 4 & 0 \end{array}\right]$ - 2X = $\left[\begin{array}{rr} 2 & 3 \\ 4 & 0 \end{array}\right]$ - 2 $\left[\begin{array}{rr} 2 / 5 & -12 / 5 \\ -11 / 5 & 3 \end{array}\right]$
= $\left[\begin{array}{rr} 2-\frac{4}{5} & 3+\frac{24}{5} \\ 4+\frac{22}{5} & 0-6 \end{array}\right]$ = $\left[\begin{array}{rr} 6 / 5 & 39 / 5 \\ 42 / 5 & -6 \end{array}\right]$
$\therefore$ Y = $\frac{1}{3}$$ \left[\begin{array}{rr} 6 / 5 & 39 / 5 \\ 42 / 5 & -6 \end{array}\right]$ = $\left[\begin{array}{rr} 2 / 5 & 13 / 5 \\ 14 / 5 & -2 \end{array}\right]$
2X + 3Y = $ \left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$ ...(i)
तथा 3X + 2Y = $\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right]$...(ii)
समी (i) को 2 से तथा समी (ii) को 3 से गुणा करके परस्पर घटाने पर,
2(2X + 3Y) - 3(3X + 2Y) = 2$\left[\begin{array}{ll} 2 & 3 \\ 4 & 0 \end{array}\right]$ - 3$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 5 \end{array}\right]$
$ \Rightarrow $ 4X + 6Y - 9X - 6Y = $\left[\begin{array}{ll} 4 & 6 \\ 8 & 0 \end{array}\right]$ - $\left[\begin{array}{rr} 6 & -6 \\ -3 & 15 \end{array}\right]$
$ \Rightarrow $ - 5X = $\left[\begin{array}{ll} 4-6 & 6+6 \\ 8+3 & 0-15 \end{array}\right]$= $\left[\begin{array}{rr} -2 & 12 \\ 11 & -15 \end{array}\right] $
$ \Rightarrow $ X = - $ \frac{1}{5}$ $\left[\begin{array}{rr} -2 & 12 \\ 11 & -15 \end{array}\right]$ = $ \left[\begin{array}{rr} 2 / 5 & -12 / 5 \\ -11 / 5 & 3 \end{array}\right] $
तब समी (i) से, 3Y = $ \left[\begin{array}{ll} 2 & 3 \\ 4 & 0 \end{array}\right]$ - 2X = $\left[\begin{array}{rr} 2 & 3 \\ 4 & 0 \end{array}\right]$ - 2 $\left[\begin{array}{rr} 2 / 5 & -12 / 5 \\ -11 / 5 & 3 \end{array}\right]$
= $\left[\begin{array}{rr} 2-\frac{4}{5} & 3+\frac{24}{5} \\ 4+\frac{22}{5} & 0-6 \end{array}\right]$ = $\left[\begin{array}{rr} 6 / 5 & 39 / 5 \\ 42 / 5 & -6 \end{array}\right]$
$\therefore$ Y = $\frac{1}{3}$$ \left[\begin{array}{rr} 6 / 5 & 39 / 5 \\ 42 / 5 & -6 \end{array}\right]$ = $\left[\begin{array}{rr} 2 / 5 & 13 / 5 \\ 14 / 5 & -2 \end{array}\right]$
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