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  4. If x + y + z = 4, 2x + 5y - 2z = 3, x + 7y - 7z = 5, then z =

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Q. If $x + y + z = 4$, $2x + 5y - 2z = 3$, $x + 7y - 7z = 5$, then $z =$

Determinants

Solution:

The given system of equations can be writtenin matrix form as
$\left[\begin{matrix}1&1&1\\ 2&5&-2\\ 1&7&-7\end{matrix}\right]\left[\begin{matrix}x\\ y\\ z\end{matrix}\right]=\left[\begin{matrix}4\\ 3\\ 5\end{matrix}\right]$
or $AX=B$, where
$A=\left[\begin{matrix}1&1&1\\ 2&5&-2\\ 1&7&-7\end{matrix}\right], X= \left[\begin{matrix}x\\ y\\ z\end{matrix}\right] and B=\left[\begin{matrix}4\\ 3\\ 5\end{matrix}\right]$

Now, $\left|A\right|=\left|\begin{matrix}1&1&1\\ 2&5&-2\\ 1&7&-7\end{matrix}\right|=-21+12+9=0$

Cofactor matrix of $A =\left[\begin{matrix}-21&12&9\\ 14&-8&-6\\ -7&4&3\end{matrix}\right]$

$\therefore \quad$ adj A $=\left[\begin{matrix}-21&12&9\\ 14&-8&-6\\ -7&4&3\end{matrix}\right]^{T}=\left[\begin{matrix}-21&14&-7\\ 12&-8&4\\ 9&-6&3\end{matrix}\right]$

Now, $\left(adj A\right) \cdot B =\left[\begin{matrix}-21&14&-7\\ 12&-8&4\\ 9&-6&3\end{matrix}\right]\left[\begin{matrix}4\\ 3\\ 5\end{matrix}\right]$

$\left[\begin{matrix}-84+42-35\\ 48-24+20\\ 36-18+15\end{matrix}\right]=\left[\begin{matrix}-77\\ 44\\ 33\end{matrix}\right]\ne O$

$\therefore \quad$ The given system of equations has no solution.