Question

Lee is thinking of two Numbers,the product is 18. the quotient is 2

Answer 1

$\begin{cases} x*y=18 \\ \frac{x}{y}=2 \end{cases}\\\\\begin{cases} 2y*y=18 \\ x=2y \end{cases}\\\\ \begin{cases} 2y^2=18\ \ /:2 \\ x=2y \end{cases}\\\\\begin{cases} y^2=9 \\ x=2y \end{cases}\\\\ y^2-9=0\\\\(y-3)(y+3)=0\\y-3=0\ \ \vee \ \ y+3=0\\y=3 \ \ \vee \ \ y=-3$

$\begin{cases} y=3 \\ x=2y \end{cases}\ \ \vee \ \ \ \ \begin{cases} y=-3 \\ x=2y \end{cases}\\\\ \begin{cases} y=3 \\ x=2*3 \end{cases}\ \ \vee \ \ \ \ \begin{cases} y=-3 \\ x=2*(-3) \end{cases}\\\\ \begin{cases} y=3 \\ x=6 \end{cases}\ \ \vee \ \ \ \ \begin{cases} y=-3 \\ x=-6\end{cases}$

Answer 2

X, y - the numbers

The product is 18.
$xy=18$

The quotient is 2.
$\frac{x}{y}=2 \ \ \ |\times y \\ x=2y$

Substitute 2y for x in the 1st equation:
$2y \times y=18 \\ 2y^2=18 \ \ \ |\div 2 \\ y^2=9 \\ y=\pm \sqrt{9} \\ y=3 \ \lor \ y=-3 \\ \\ x=2y \\ x=2 \times 3 \ \lor \ x=2 \times (-3) \\ x=6 \ \lor \ x=-6 \\ \\ (x,y)=(6,3) \hbox{ or } (x,y)=(-6,-3)$

The numbers are 6 and 3 or -6 and -3.