Express $z$ in terms of $x$ and $y$, i.e., find $z= f(x,y)$

Question

I've been banging my head against the wall for a while now:

$x = s^2 - t^2$

$y = s + t$

$z = s^2 + 3t$

Express $z$ in terms of $x$ and $y$.

Answer 1

Hint: What is $\dfrac{x}{y}$ equal to?

Answer 2

Try to express $s$ and $t$ as functions of $x$ and $y$ from the first two equations. Then plug these expressions into the third equation.

Answer 3

$$x=s^{2}-t^{2}=(s+t)(s-t)$$ so $$s+t=\frac{x}{s-t}$$ $$s-t=\frac{x}{s+t}=\frac{x}{y}$$

$$(s+t) + (s-t) = 2s=\frac{x}{s-t}+\frac{x}{y}$$ $y=s+t$, so $t=y-s$ and therefore: $$2s=\frac{x}{2s-y}+\frac{x}{y}=x(\frac{1}{2s-y}+\frac{1}{y})$$ $$2s=x(\frac{y}{2sy-y^2}+\frac{2s-y}{2sy-y^2})=x(\frac{2s}{2sy-y^2})$$ $$1=\frac{x}{2sy-y^2}$$ $$2sy-y^2=x$$ $$2sy=x+y^2$$ $$s=\frac{x+y^2}{2y}$$ From $z=s^2+3t$ we have: $$z=(\frac{x+y^2}{2y})^{2}+3t$$ $y=s+t$ so $t=y-\frac{x+y^2}{2y}$ and finally: $$z=(\frac{x+y^2}{2y})^{2}+3(y-\frac{x+y^2}{2y})$$ Pretty sure this is correct...