Let's solve the equation [tex]\(27x = \frac{9}{3x}\)[/tex] step-by-step.
1. Start with the given equation:
[tex]\[
27x = \frac{9}{3x}
\][/tex]
2. Clear the fraction by multiplying both sides of the equation by [tex]\(3x\)[/tex]:
[tex]\[
27x \cdot 3x = \frac{9}{3x} \cdot 3x
\][/tex]
3. Simplify both sides:
[tex]\[
81x^2 = 9
\][/tex]
4. Divide both sides by 9 to isolate the [tex]\(x^2\)[/tex] term:
[tex]\[
81x^2 \div 9 = 9 \div 9
\][/tex]
[tex]\[
9x2 = 1
\][/tex]
5. Divide both sides by 9 to further simplify:
[tex]\[
x^2 = \frac{1}{9}
\][/tex]
6. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{1}{9}}
\][/tex]
7. Simplify the square root:
[tex]\[
x = \pm \frac{1}{3}
\][/tex]
So, the solutions to the equation [tex]\(27x = \frac{9}{3x}\)[/tex] are [tex]\(x = \frac{1}{3}\)[/tex] and [tex]\(x = -\frac{1}{3}\)[/tex].
### Conclusion
The correct solutions are:
[tex]\[
x = \frac{1}{3} \quad \text{and} \quad x = -\frac{1}{3}
\][/tex]
These are neither [tex]\(0\)[/tex] nor [tex]\(\frac{1}{2}\)[/tex], so the options given (a) and (b) are incorrect.
