Determine all real (a,b,c) that satisfy the systems of equations.

Question

Determine all real (a,b,c) that satisfy the equations

$a^2b^3c=-(2^7)$

$ab^2c^3=(2^8)$

$a^3bc^2=-(2^6)$

I have tried making every variable a base 2 (and (-2)) to an exponent and then use systems of equations to solve for the exponents. For example, $2u+3v+w=7$. This method is not working since I can't account for the negative values.

Answer 1

First determine the signs of the variables. The second equation says $a$ and $c$ have the same sign. The third says $a$ and $b$ have opposite signs and the first says $b$ and $c$ have opposite signs. That gives two possibilities: $a,c \gt 0, b\lt 0$ and $a,c \lt 0, b \gt 0$. We can take the absolute values of everything and apply the signs at the end.

Now taking base $2$ logs is a good thing to do. Letting $A=\log_2(|a|), B=\log_2(|b|), C=\log_2(|c|)$ we have $$2A+3B+C=7\\A+2B+3C=8\\3A+B+2C=6$$ which we can solve with the usual techniques, getting $$A=\frac 12,B=C=\frac 32$$ so the solutions are $$a=\sqrt 2,b=-2\sqrt 2, c=2\sqrt 2\\a=-\sqrt 2,b=2\sqrt 2, c=-2\sqrt 2$$

Answer 2

Hint: Try multiplying the three equations together.