Question

If we have an expression as \[{a^{x - 1}} = bc,{b^{y - 1}} = ac,{c^{z - 1}} = ab\], then find the value of \[xy + yz + zx - xyz\].

Answer 1

Hint: In the given question, we have been given an algebraic expression. This expression involves the use of various variables raised to the power of some other variables. We have to find the value of the given expression. To solve that, we are going to use the formula of product and division of powers to bring down the powers. Then we are going to simplify the value of the variables in terms of the given variables. And finally, we are going to substitute the values and find our answer.

Complete step by step solution:
We have been given, \[{a^{x - 1}} = bc,{b^{y - 1}} = ac,{c^{z - 1}} = ab\].
We know, \[{k^{m - n}} = \dfrac{{{k^m}}}{{{k^n}}}\]
So, \[{a^{x - 1}} = \dfrac{{{a^x}}}{a} = bc \Rightarrow {a^x} = abc \Rightarrow a = {\left( {abc} \right)^{\dfrac{1}{x}}}\]
Similarly, \[b = {\left( {abc} \right)^{\dfrac{1}{y}}}\]
And finally, \[c = {\left( {abc} \right)^{\dfrac{1}{z}}}\]
Now, multiplying the three terms together,
\[abc = {\left( {abc} \right)^{\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}}}\]
Now, as the bases are same, we can equate the powers, so, we have,
\[1 = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}\]
Taking the LCM and solving,
\[1 = \dfrac{{yz + xz + xy}}{{xyz}}\]
Taking the denominator to the other side,
\[xyz = xz + yz + zx\]
Hence, \[xz + yz + zx - xyz = 0\]

Note: In the given question, we were given an algebraic expression. This algebraic expression had some variables which had powers which were also variables. We had to find the value of the expression given in the question. We solved that by simplifying the value of the variables in terms of the given variables. Then we put in the values into the expression, simplified them and found our answer.