How to solve these $3$ equations for three unknowns $x$,$y$,$z$? [duplicate]

Hint: Put $$X=x+1$$ $$Y=y+1$$ $$Z=z+1$$

Then we have

$$XY=24$$ $$YZ=32$$ $$ZX=48$$

Can you take it from there?

We can use Simon's favorite factoring trick.

$$xy+x+y+1=(x+1)(y+1)$$

This tells us

$$(x+1)(y+1)=24$$$$(y+1)(z+1)=32$$$$(x+1)(z+1)=48$$So, we know that $x+1 = \pm\frac{\sqrt{24\cdot32\cdot48}}{32}\to x=5,-7$. Likewise, you can find the other variables.

You can convert the first equation into an equation that expresses y in terms of x.

You can convert the third equation into an equation that expresses z in terms of x.

You can substitute these formulas for y and z into the second equation. This gives a single equation, in a single variable (x).

The single equation can be simplified, by letting v = x + 1, and substituting v-1 in for x. Then you can multiply through by v². Notice that you are assuming that v ≠ 0. Then you can solve for v. Since this is a second order equation, you will get two solutions (which might be equal to each other).

Now you can solve for x. (It is v-1). Notice that you are assuming that x ≠ -1.

Now you can solve the first equation for y, and the third equation for z.

Now you need to make sure you don't have a divide-by-zero error. In other words, check what values you get for y and z if x were equal to -1. Since these are the asymptotes of the hyperbolas that are hinted at by marshal craft's answer, y and z turn out to be ±∞. This demonstrates that it was okay to assume that x ≠ -1.

Now you can perform your check-by-substitution, to verify that both solutions are correct.