This is not a homework question; rather a review for a Mechanical Engineering Board Exam. I need to find an efficient way to solve equations of these types:
(x+y)(x+y+z) = 384
(y+z)(x+y+z) = 288
(x+z)(x+y+z) = 480
It has been showing up a bit on my reviewers. I have no idea how to go about it. We're only allowed an fx-991es plus Casio Calculator. Thanks
Beside the good hint Alexey Burdin gave in comments, add an extra variable $t=x+y+z$. So, now, you have four equations for four unknowns $$(x + y) t=384$$ $$(y + z) t=288$$ $$ (x + z) t=480$$ $$t=x+y+z$$ From the first three eliminate $x$, $y$ and $z$ as functions of $t$ (this is simple since the equations are now linear assuming that you know $t$).
Doing so, you get $$x= \frac{288}{t}\, ,\,y= \frac{96}{t}\,,\,z= \frac{192}{t} $$ Now, replace these expressions in the last equation; this leads to $$t=\frac{576}{t}$$ that is to say $t=\pm 24$. Report this value in the last expressions of $x,y,z$ as functions of $t$ to get the results.