To solve the equation \( 6^{2x} = 6^{1 - 3x} \), we can make use of the property of exponents that states if the bases are the same, then the exponents must be equal. 

Given the equation:

\[
6^{2x} = 6^{1 - 3x}
\]

We can equate the exponents:

\[
2x = 1 - 3x
\]

Now, solve for \( x \):

1. Add \( 3x \) to both sides to get all the \( x \) terms on one side:

\[
2x + 3x = 1
\]

\[
5x = 1
\]

2. Divide both sides by 5:

\[
x = \frac{1}{5}
\]

So, the solution is:

\[
x = \frac{1}{5}
\]

Would you like any details or have any questions about this solution? 

Here are five related questions:

1. How would you solve \( 7^{3x} = 7^{2x + 4} \)?
2. What happens if the bases are different, like in \( 5^{2x} = 7^{1 - x} \)?
3. Can you solve \( 6^{2x + 1} = 6^{3x - 2} \)?
4. What is the value of \( x \) in \( 9^{4x} = 9^{x + 6} \)?
5. How would you solve \( 10^{3x} = 10^{2 - 2x} \)?

**Tip:** When solving exponential equations with the same base, you can equate the exponents directly. This simplifies the process and helps you quickly find the value of the variable.

Explore how to solve the exponential equation 6^(2x) = 6^(1 - 3x) by equating exponents and applying basic algebraic techniques. This detailed explanation covers the laws of exponents and simplifies the problem to find x = 1/5. Ideal for students in Grades 9-10.

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