91Ó°ÊÓ

The weight of metalscan be found using the augmented matrix.

The total weight of crown isminae.

Let G = Gold, B= Bronze, T =Tin, I =Iron

Therefore, according to question

$G+B+T+I=60.......\left(1\right)$

Gold and bronze makes up$\frac{2}{3}$ weight of the crown.

$$\begin{gathered} G+B=\frac{2}{3}×60 \\ ∴G+B=40......\left(2\right) \end{gathered}$$

Gold and tin makes up$\frac{3}{4}$weight of the crown.

$$\begin{gathered} G+T=\frac{3}{4}×60 \\ ∴G+T=45.....\left(3\right) \end{gathered}$$

Gold and iron makes up$\frac{3}{5}$weight of the crown.

$$\begin{gathered} G+I=\frac{3}{5}×60 \\ ∴G+I=36.......\left(4\right) \end{gathered}$$

Consider the equations (1), (2)(3)and (4). The system of equations is,

$\left|\begin{array}{c}G+B+T+I=60\\ G+B=40\\ G+T=45\\ G+I=36\end{array}\right|$

Represent the above equations in terms of a matrix.

$\left|\begin{array}{ccccc}1& 1& 1& 1& 60\\ 1& 1& 0& 0& 40\\ 1& 0& 1& 0& 45\\ 1& 0& 0& 1& 36\end{array}\right|$

Consider the matrix.

$$\begin{gathered} \left|\begin{array}{ccccc}1& 1& 1& 1& 60\\ 1& 1& 0& 0& 40\\ 1& 0& 1& 0& 45\\ 1& 0& 0& 1& 36\end{array}\right| \\ =\left|\begin{array}{ccccc}1& 1& 1& 1& 60\\ 0& -1& 0& -1& -15\\ 0& 0& -1& -1& -20\\ 0& 0& 0& 2& 11\end{array}\right| \\ =\left|\begin{array}{ccccc}1& 0& 0& 0& \frac{61}{2}\\ 0& 1& 0& 0& \frac{19}{2}\\ 0& 0& 1& 0& \frac{29}{2}\\ 0& 0& 0& 1& \frac{11}{2}\end{array}\right| \end{gathered}$$

The values are, $G=\frac{61}{2}=30.5,B=\frac{19}{2}=9.5,T=\frac{29}{2}=14.5,I=\frac{11}{2}=5.5$

The metal gold should weigh 30.5 minae, the metal bronze should weigh 9.5 minae, the metal tin should weigh 14.5 minae and the metal iron should weigh 5.5 minae.