Question

Find the ratios of $x:y:z$ from the equations
$7x=4y+8z,3z=12x+11y$.

Answer 1

By transposition we have $7x-4y-8z=0,12x+11y-3z=0$.

Write down the coefficients, thus

$x=\left|\begin{array}{cc}-4& -8\\ 11& -3\end{array}\right|$

$y=\left|\begin{array}{cc}-8& 7\\ -3& 12\end{array}\right|$

$z=\left|\begin{array}{cc}7& -4\\ 12& 11\end{array}\right|$

whence we obtain the products

$(-4)\times (-3)-11\times (-8),(-8)\times 12-(-3)\times 7,7\times 11\times -12\times (-4),$

or $100,-75,125$;

$\therefore \frac{x}{100}=\frac{y}{-75}=\frac{z}{125},$

that is , $\frac{x}{4}=\frac{y}{-3}=\frac{z}{5}.$