Multivariable linear equation with 3 variables

Question

How can I solve this system of equations of variables x, y and z: $$ xy -2 \sqrt y + 3yz = 8 \\ 2xy -3 \sqrt y + 2yz = 7\\ -xy + \sqrt y + 2yz = 4 $$

I'm used to solve problems with singular variables ( like 2x +3y-5z= k), and I saw this problem on an exam I want to aply. Thanks for giving me at least one idea how to solve it.

Answer 1

Let $u=xy$, $v=\sqrt y$, and $w=yz$. Solve for $u,\ v$, and $w$. Then, solve for your original variables, as you'll have: $$\begin{align} xy&=u\\ \sqrt y&=v\\ yz&=w \end{align} $$ and you'll know the values of the three variables on the right. From there, solve for $y$ first. and use substitution to solve the rest.

Answer 2

Can you solve the following linear system linear system in $a_1,a_2,a_3 \in \mathbb{R}$? \begin{align*} a_1-2a_2+3a_3 &= 8 \\ 2a_1 - 3a_2 + 2a_3 &= 7 \\ -a_1 + a_2 + 2a_3 &= 4 \end{align*}

Answer 3

The resolution of the "linear" system leads to a solution of the form

$$\begin{cases}xy&=a,\\\sqrt y&=b,\\yz&=c.\end{cases}$$

Now taking the logarithm and using uppercase to denote it, this is equivalent to

$$\begin{cases}X+Y&=A,\\\dfrac12Y&=B,\\Y+Z&=C,\end{cases}$$ another linear system.