how to solve these kind of systems $x^2+y^2=z^2; z-y^3=5; xy=z$

Question

Three variable system of equations with three variable with exponents for example

$x^2+y^2=z^2$

$z-y^3=5$

$xz=y$

Answer 1

Basically, you can simplify this set of equations by substituting. For example, in this example, you can first divide both sides of the third equation by $z$, because $z\neq0$, or else $y=0$ and you get $0-0=5$, which is a contradiction. After dividing both sides of the third equation by $z$, you would get $x=\frac{y}{z}$, and after substituting that into the first equation, you would get

$$\left(\frac{y}{z}\right)^2+y^2=z^2$$ $$\frac{y^2}{z^2}+y^2=z^2$$

Combining this with the second equation after using the same method again, you eventually can get the answer.

But you can also use a very powerful mathematical calculator, such as Wolfram|Alpha. The link to that is https://www.wolframalpha.com/input?i2d=true&i=Power%5Bx%2C2%5D%2BPower%5By%2C2%5D%3DPower%5Bz%2C2%5D%5C%2844%29+Power%5Bz%2C3%5D-y%3D5%5C%2844%29+xz%3Dy, which is the specific search for this set of three equations.