Examine the consistency of the system of equa | vedclass

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(A) The given system of equations is:$x+y+z=1$$2x+3y+2z=2$$ax+ay+2az=4$The system can be written as $AX=B$,where$A = \begin{bmatrix} 1 & 1 & 1 \ 2 &

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Consistent for all $a 
eq 0$

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(A) The given system of equations is:$x+y+z=1$$2x+3y+2z=2$$ax+ay+2az=4$The system can be written as $AX=B$,where$A = \begin{bmatrix} 1 & 1 & 1 \ 2 & 3 & 2 \ a & a & 2a \end{bmatrix}$,$X = \begin{bmatrix} x \ y \ z \end{bmatrix}$,and $B = \begin{bmatrix} 1 \ 2 \ 4 \end{bmatrix}$.Calculate the determinant $|A|$:$|A| = 1(6a - 2a) - 1(4a - 2a) + 1(2a - 3a)$$|A| = 4a - 2a - a = a$.Case $1$: If $a 
eq 0$,then $|A| 
eq 0$. The matrix $A$ is non-singular,so the system has a unique solution and is consistent.Case $2$: If $a = 0$,the equations become:$x+y+z=1$$2x+3y+2z=2$$0=4$Since $0=4$ is a contradiction,the system is inconsistent when $a=0$.Thus,the system is consistent for all $a 
eq 0$.

Examine the consistency of the system of equations: $x+y+z=1$,$2x+3y+2z=2$,and $ax+ay+2az=4$.

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