#### Question

If a , b , c are real numbers such that 3^{a+b-c}7^{2b+c}5^{2a-b+c}=27 , find the value of a , b, c

Question

If a , b , c are real numbers such that 3^{a+b-c}7^{2b+c}5^{2a-b+c}=27 , find the value of a , b, c

To make the equation hold true

3^{a+b-c}7^{2b+c}5^{2a-b+c}=27

the powers of 3, 7 and 5 must meet the following conditions.

Hence, a system of equations in terms of a,b, c is obtained

\begin{cases} a+b-c = 3 \\ 2b+c = 0 \\ 2a-b+c=0 \end{cases}

Solving the system of equations yields the values of a , b, c

\begin{cases} a &= 1 \\ b&=\dfrac{2}{3} \\ c&=-2b=-\dfrac{4}{3} \end{cases}