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# Find the values of x,y and z from the following equations:$(i)\begin{bmatrix}4 & 3\\x & 5\end{bmatrix}=\begin{bmatrix}y & z\\1 & 5\end{bmatrix}\qquad(ii)\;\begin{bmatrix}x+y & z\\5+z & xy\end{bmatrix}=\begin{bmatrix}6 & 2\\5 & 8\end{bmatrix}\qquad(iii)\;\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}$ Comment
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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.
(i)$\begin{bmatrix}4 & 3\\x & 5\end{bmatrix}=\begin{bmatrix}y & z\\1 & 5\end{bmatrix}.$
The above given matrices are equal.Hence we can obtain the value of x,y,z by comparing the matrices corresponding elements.
By comparing we get
$\Rightarrow 4=y \;z=3\;x=1.$
(ii)$\begin{bmatrix}x+y & 2\\5+z & xy\end{bmatrix}=\begin{bmatrix}6 & 2\\5 & 8\end{bmatrix}.$
The above given matrices are equal.Hence we can obtain the value of x,y,z can be obtained by comparing the matrices.
$\Rightarrow$ since the matrices are equal their corresponding elements should be equal.
x+y=6------(i)
5+z=5-----(ii)
xy=8------(iii)
Consider equation (i) & (iii)&simplifying
x=4,y=2 or x=2,y=4.
From equ(iii)xy=8
$x=\frac{8}{y}$
Substitute the value of x in equation(i)
$\frac{8}{y}+y=6.$
$\frac{8+y^2}{y}=6\Rightarrow 8+y^2=6y.$
$y^2-6y+8=0$
$y^2-4y-2y+8=0.$
(y-4)(y-2)=0
y=4,y=2.
Substitute the value of y in equ(iii)
If y=4,
x(4)=8$\Rightarrow x=2.$
If y=2,
x(2)=8$\Rightarrow x=4.$
From equ(iii)
5+z=5
z=5-5=0.
(iii)$\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}.$
Given matrices are equal hence their corresponding elements should be equal.
x+y+z=9----(1)
x+z=5------(2)
y+z=7------(3)
Consider equations (1)&(2)
Substitute the value of (2) in eq(1)
x+y+z=9
y+5=9
y=9-5
y=4.
Substitute the value of y in eq(3)
y+z=7
4+z=7
z=3.
Substitute the value of z in eq(2)
x+z=5
x+3=5.
x=2.