Question

$x+2y+z=4$
$2x+y+2z=5$
$x-y+z=1$

The system of algebraic equations given above has

• A. a unique solution of$x=1,y=1andz=1$
• B. only the two solutions of $(x=1,y=1andz=1)and(x=2,y=1andz=0)$
• C. infinite number of solutions
• D. no feasible solution

Answer 1

The correct option is C infinite number of solutions

The augmented matrix for the given system of algebraic equation is given by

$[A:B]=\left[\begin{array}{cccc}1& 2& 1& :4\\ 2& 1& 2& :5\\ 1& -1& 1& :1\end{array}\right]$
$=\left[\begin{array}{cccc}1& 2& 1& :4\\ 0& -3& 0& :-3\\ 0& -3& 0& :-3\end{array}\right];\begin{array}{c}{R}_3\to R_3-R_1\\ R_2\to R_2-2R_1\end{array}$
$=\left[\begin{array}{cccc}1& 2& 1& :4\\ 0& -3& 0& :-3\\ 0& 0& 0& :0\end{array}\right];R_3\to R_3-R_2$

$\therefore Rank[A:B]=$ Rank [A] = 2 < number of unknowns (3)

$\therefore$ The system, is consistent and has infinite solutions.