Find the values of x,y and z, if $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$

Toolbox:
• If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij}=B_{ij}$, then any element $a_{ij}$ in matrix A is equal to corresponding element $b_{ij}$ in matrix B.
• We can then match the corresponding elements and solve the resulting equations to find the values of the unknown variables.
Step1:
Given:
$\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$
The given two matrices are equal ,hence their corresponding elements should be equal.
$\Rightarrow x+y+z=9$-----(1)
$\;\;\;x+z=5$-------(2)
$\;\;\;y+z=7$-------(3)
From equation (2) we have
$z=5-x$ or $x=5-z$
From equation (3) we have
$y+z=7$
$y=7-z$
Step2:
Replace the value of x & y in equation (1)
$x+y+z=9$
$(5-z)+(7-z)+z=9$
$5-z+7-z+z=9$
$-z+12$=9
$-z=9-12$
$-z=-3$
$z=3$
Step3:
We know that $x=5-z$
$x=5-3$
$x=2$
We know that $y=7-z$
$y=7-3$
y=4.
Hence x=2,y=4,z=3.