Examine the consistency of the system of equations
`x + y + z = 1`,`2x + 3y + 2z = 2`,`a x + a y + 2a z = 4`

Explanation

Simplifying `3^(rd)` equation

`a x + a y + 2a z = 4`

`a( x + y + 2 z) = 4`

` x + y + 2 z = (4)/(a)`

Now system of equation is

`x + y + z = 1`

`2x + 3y + 2z = 2`

` x + y + 2 z = (4)/(a)`

Writing equation as `AX=B`

`[[1,1,1],[2,3,2],[1,1,2]][[x],[y],[z]]=[[1],[2],[4/a]]`

Here,
`A=[[1,1,1],[2,3,2],[1,1,2]]`, `X=[[x],[y],[z]]`

`B=[[1],[2],[4/a]]`


Calculating `|A|`

`|A|=|[1,1,1],[2,3,2],[1,1,2]|`

` " " " " =1[[3,2],[1,2]]-1[[2,2],[1,2]]+1[[2,3],[1,1]]`

` " " " " =1(6-2)-1(4-2)+1(2-3)=4-2-1`

` " " " " =1`


Since `|A|ne0`
Hence, system of equation is consistent.