The system of equations

x + y + z = 150

x + 2y + 3z = 100

2x + 3y + 4z = 200   has

Explanation:

By using Gaussian elimination method:

\(\begin{array}{*{20}{c}} {{R_1}}\\ {{R_2}}\\ {{R_3}} \end{array}\left[ {\begin{array}{*{20}{c}} 1&1&1\\ 1&2&3\\ 2&3&4 \end{array}\left| {\begin{array}{*{20}{c}} {150}\\ {100}\\ {200} \end{array}} \right.} \right]\)

\(\Rightarrow {R_2} - {R_1}\ {\rm{and}}\ {R_3} - 2{R_1} \Rightarrow\)

\(\begin{array}{l} = \left[ {\begin{array}{*{20}{c}} 1&1&1\\ 0&1&2\\ 0&1&2 \end{array}\left| {\begin{array}{*{20}{c}} {150}\\ { - 50}\\ { - 100} \end{array}} \right.} \right]\\ \Rightarrow {R_3} - {R_2}\\ \left[ {\begin{array}{*{20}{c}} 1&1&1\\ 0&1&2\\ 0&0&0 \end{array}\left| {\begin{array}{*{20}{c}} {150}\\ { - 50}\\ { - 50} \end{array}} \right.} \right] \end{array}\)

1x + 1y + 1z = 150

1y + 2z = -50

0 = -50

Here, two equations and 3 unknowns as third equation is invalid

and also rank of [A] < rank of [A : B]

So inconsistent and having no solution