Find the value of 12x/y

Question

Given that $6x^2 - xy = 7y^2,$ find the value of $12x/y$.

I am totally stuck and can even find a way out.

Thanks in advance.

Answer 1

If you divide the equation by $y^2$ you get a quadratic in $\frac xy$. You should note that $x=0,y=0$ is one solution to your equation, but presumably you are being asked for another because $\frac xy$ doesn't make sense when $y=0$

Answer 2

Let $y=mx$, then $$7m^2+m-6=0 \Rightarrow m=-1, 6/7.$$ So $$12 x/y=-12~ or ~ 14.$$

Answer 3

You may also proceed as follows:

• $6x^2 - xy -7y^2= (6x-7y)(x+y) \stackrel{!}{=}0$

Not considering the case $x=y=0$ you get

• $6x-7y= 0 \Rightarrow \boxed{12\frac{x}{y} = 14}$ or
• $x+y = 0 \Rightarrow \boxed{12\frac{x}{y} = -12}$