I need help getting this equation solved for a website I am building. I am pretty bad at math and am only in pre-algebra. I don't know how I would go about canceling out the ^5 and ^4 because I can't square-root it. Thank you for your help.
$$x y^5=8000$$
and
$$x y^4>4100$$
On
First consider if the two variables are both positive, both negative or one positive, one negative.
It should be clear from the first equation that either both have to be positive or both negative.
From the second inequality, both have to be positive.
Hence you're only considering the case where $x, y > 0$.
In this case, it's possible to divide the equation by the inequality, flipping the direction, giving $y < \frac{80}{41}$.
From the first equation, $x = \frac{8000}{y^5}$
That's basically the solution. $x$ will remain dependent on $y$ because of the inequality.
On
From the first equation $x = \frac{8000}{y^5}$, plugging into the inequality, you have:
$$\left(\frac{8000}{y^5} \right)y^4 > 4100$$
$$\frac{8000}{y} > 4100$$
If $y < 0$, the inequality does not hold. If $y>0$, you have $y < \frac{8000}{4100} = \frac{80}{41}$. So the equation and inequality are both satisfied only for points:
$$(x,y) = \left( \frac{8000}{y^5}, y\right), \text{ where } 0 < y < \frac{80}{41}$$
On
One thing you could do is to consider two equations $$x y^5=8000$$ $$x y^4=4100+a$$ where $a$ is a positive number in order to satisfy the inequality. Dividing the first by the second gives the solution $$y=\frac{8000}{a+4100}$$ which, replaced in the first equation, leads to $$x=\frac{(a+4100)^5}{8000^4}$$. These show how the results depend on the value assigned to $a$.
For your problem, I would solve the equation for one of the variables and plug it into the inequality.
Actually, I would take a shortcut: multiply the second equation by something that makes substitutino easier. (but don't forget that, for example, you would need to treat the $y>0$ and $y<0$ cases separately if you multiply an inequation by $y$)