To solve the equation [tex]\(6^{3-x} = 6^2\)[/tex], let's follow the steps in detail:
1. Identify the Equation:
The given equation is [tex]\(6^{3-x} = 6^2\)[/tex].
2. Equate the Exponents:
Since the bases (6) on both sides of the equation are the same, we can set the exponents equal to each other. This is a property of exponents when the bases are identical:
[tex]\[
3 - x = 2
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, we simply solve the linear equation [tex]\(3 - x = 2\)[/tex].
- Start by isolating [tex]\(x\)[/tex]. Subtract 3 from both sides:
[tex]\[
3 - x - 3 = 2 - 3
\][/tex]
Simplifying this gives:
[tex]\[
-x = -1
\][/tex]
- Multiply both sides by [tex]\(-1\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
-1 \times -x = -1 \times -1
\][/tex]
Simplifying this gives:
[tex]\[
x = 1
\][/tex]
4. Conclusion:
The solution to the equation [tex]\(6^{3-x} = 6^2\)[/tex] is [tex]\(x = 1\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
Among the given options:
A. [tex]\(x = 9\)[/tex]
B. [tex]\(x = 1\)[/tex]
C. [tex]\(x = -1\)[/tex]
D. [tex]\(x = -6\)[/tex]
The correct answer is [tex]\(x = 1\)[/tex].
