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# Find the values of x,y and z from the following equations: $(iii)\;\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}$

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.
Given $\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}.$ Since the order matrices are equal Since these matrices are equal, we can an obtain the value of x,y,z by comparing the matrices' corresponding elements.
By comparing the given two matrices of equal order, we can see that:
$x+y+z = 9$ (i)
$x+z = 4$ (ii)
$y+z=7$ (iii)
From (iii) we get $z=7-y$.
Substituting for $z$ in (i), we get $x+y+7-y=9$ $\rightarrow$ $x+7 = 9$ $\rightarrow$ $x=2$.
Substituting $x=2$ in (ii), we get $2+z = 5 \rightarrow$ $z=3$.
Substituting $z=4$ in (iii), we get $y+3 = 7 \rightarrow y=4$.
Solving for x, y and z we get $(x,y,z) = (2,3,4)$.