How can I solve an equation system where two variables are multiplied by each other on a TI-89? For example:
$$\begin{cases} 30 - Z + 0.02 \cdot Y \cdot Z = 0\\ 50 - Z + 0.02 \cdot X \cdot Z = 0\\ 50 - 0.02 \cdot X \cdot Y - X - Y = 0 \end{cases} $$
I was searching and I couldn't find anything, is it possible? Any help would be great, thanks!
On
We have
$$30 - Z + 0.02\cdot Y \cdot Z = 0 \tag{1}$$
$$50 - Z + 0.02 \cdot X\cdot Z = 0 \tag{2}$$
$$50 - 0.02 \cdot X \cdot Y - X - Y = 0 \tag{3}$$
By $(1)$ and $(2)$ we have $$Y=\frac{50(Z-30)}Z$$ and $$X=\frac{50(Z-50)}Z$$
Then, substituting into $(3)$ we have
$$50 - 0.02 \cdot \bigg( \frac{50(Z-50)}Z \bigg) \cdot \bigg( \frac {50(Z-30)}Z \bigg)- \bigg( \frac{50(Z-50)}Z\bigg) -\bigg( \frac {50(Z-30)}Z\bigg) = 0$$ which, upon simplifying, gives
$$Z^2 - 80 Z + 750 = 0 \tag{for Z≠0}$$
Using the quadratic formula we get $$Z=40 \pm 5\sqrt{34}$$
Can you take it from here?
The first equation becomes $Y=\frac {50(Z-30)}Z$ and the second $X=\frac{50(Z-50)}Z$. Plug those into the third, note that $Z\neq 0$ so you can multiply by $Z^2$ and you have a quadratic.