The system of equations x + y + z = 9, 2x + 5y + 7z = 52, 2x + y - z = 0 are consistent and
have unique solution (x, y, z) then $x - y + z =$ __________

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answer is 3.

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Detailed Solution

$\begin{array}{l} AD = [\begin{array}{llll} 1 & 1 & 1 & 9 \\ 2 & 5 & 7 & 52 \\ 2 & 1 & - 1 & 0 \end{array}], A = [\begin{matrix} 1 & 1 & 1 \\ 2 & 5 & 7 \\ 2 & 1 & - 1 \end{matrix}] \\ R_{2} \to R_{2} - 2 R_{1}, R_{3} \to R_{3} - 2 R_{1} [\begin{matrix} 1 & 1 & 1 & 9 \\ 0 & 3 & 5 & 34 \\ 0 & - 1 & - 3 & - 18 \end{matrix}] \\ R_{3} \to 3 R_{3} + R_{2} [\begin{matrix} 1 & 1 & 1 & 9 \\ 0 & 3 & 5 & 34 \\ 0 & 0 & - 4 & - 20 \end{matrix}] \end{array}$
Rank A=3, Rank(AD) =3, unknown variables =3
Costs tent and it has unsure solution
x+ y+ z=9
3y + 5z= 34
-4z =-20 $\Rightarrow$ z=5
3y+25=34 $\Rightarrow$ 3y=9 $\Rightarrow$ y=3
x+3+5=9 $\Rightarrow$ x=1
x-y+z=5-3+1=3