Question

Solve the system of equations
$x+y+z=6x+2y+3z=14x+4y+7z=30$

• A. unique solution
• B. infinite
• C. no solution
• D. multiple solutions

Answer 1

Answer: (B)

infinite

We have
$x+y+z=6x+2y+3z=14x+4y+7z=30$
The given system of equations in the matrix form are written as below:
$\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 4& 7\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6\\ 14\\ 30\end{array}\right]$
$AX=B...\left(1\right)$
where
$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 4& 7\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}6\\ 14\\ 30\end{array}\right]$
$|A|=1(14-12)-1(7-3)+1(4-2)=2-4+2=0$
$\therefore$The equation either has no solution or an infinite number of solutions.
To decide about this, we proceed to find $\left(AdjA\right)B$.
Let
$C$ be the matrix of cofactors of elements in $|A|$ such that $C=\left|\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right|$
Here
$C_{11}=\left|\begin{array}{cc}2& 3\\ 4& 7\end{array}\right|=2;C_{12}=-\left|\begin{array}{cc}1& 3\\ 1& 7\end{array}\right|=-4;C_{13}=\left|\begin{array}{cc}1& 2\\ 1& 4\end{array}\right|=2$
$C_{21}=-\left|\begin{array}{cc}1& 1\\ 4& 7\end{array}\right|=-3;C_{22}=\left|\begin{array}{cc}1& 1\\ 1& 7\end{array}\right|=6;C_{23}=-\left|\begin{array}{cc}1& 1\\ 1& 4\end{array}\right|=-3$ $C_{31}=\left|\begin{array}{cc}1& 1\\ 2& 3\end{array}\right|=1;C_{32}=-\left|\begin{array}{cc}1& 1\\ 1& 3\end{array}\right|=-2;C_{33}=\left|\begin{array}{cc}1& 1\\ 1& 2\end{array}\right|=1$
$C=\left[\begin{array}{ccc}2& -4& 2\\ -3& 6& -3\\ 1& -2& 1\end{array}\right]$
$\therefore AdjA=\left[\begin{array}{ccc}2& -3& 1\\ -4& 6& -2\\ 2& -3& 1\end{array}\right]$
Then $\left(AdjA\right)B=\left[\begin{array}{ccc}2& -3& 1\\ -4& 6& -2\\ 2& -3& 1\end{array}\right]\left[\begin{array}{c}6\\ 14\\ 30\end{array}\right]$
$=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]=O$
Hence, both conditions $|A|=0$ and $\left(AdjA\right)B=)$ are satisfied then the system of equations is consistent and has an infinite number of solutions.