Solve the following system of equations: $\;$ $x + 2y - z = 7, \;\; 2x - y + z = 2, \;\; 3x - 5y + 2z = -7$
Given system of equations:
$x + 2y - z = 7$ $\;\;\; \cdots \; (1)$
$2x - y + z = 2$ $\;\;\; \cdots \; (2)$
$3x - 5y + 2z = -7$ $\;\;\; \cdots \; (3)$
Adding equations $(1)$ and $(2)$ gives
$3x + y = 9$ $\;\;\; \cdots \; (4)$
Multiply equation $(1)$ with $2$ and add to equation $(3)$. We get
$5x - y = 7$ $\;\;\; \cdots \; (5)$
Adding equations $(4)$ and $(5)$ gives
$8x = 16$ $\implies$ $x = 2$
Substitute the value of $x$ in equation $(4)$ to get
$6 + y = 9$ $\implies$ $y = 3$
Substituting the values of $x$ and $y$ in equation $(2)$ gives
$4 - 3 + z = 2$ $\implies$ $z = 1$
$\therefore \;$ The solution to the given system of equations is $\;$ $\left(x, y, z\right) = \left\{\left(2, 3, 1\right) \right\}$