Question
Solve for \( x \). \[ \begin{array}{l}5^{(6 x-1)}=5^{(4 x+3)} \\ \text { A. } x=\frac{1}{2} \\ \text { B. } x=5 \\ \text { D. } x=2 \\ \text { D. } x=\frac{1}{5}\end{array} \]
Ask by Welch Griffiths. in the United Arab Emirates
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 2 \)
Solution
To solve the equation:
\[
5^{6x - 1} = 5^{4x + 3}
\]
Since the bases are the same and positive (\(5 \neq 1\)), we can set the exponents equal to each other:
\[
6x - 1 = 4x + 3
\]
Subtract \(4x\) from both sides:
\[
2x - 1 = 3
\]
Add \(1\) to both sides:
\[
2x = 4
\]
Divide both sides by \(2\):
\[
x = 2
\]
**Answer:**
\( \boxed{2} \)
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Simplify this solution Extra Insights
To solve the equation \( 5^{(6x-1)} = 5^{(4x+3)} \), we can set the exponents equal to each other since the bases are the same: \[ 6x - 1 = 4x + 3 \] Next, we will isolate \( x \): 1. Subtract \( 4x \) from both sides: \[ 6x - 4x - 1 = 3 \] which simplifies to \[ 2x - 1 = 3 \] 2. Add 1 to both sides: \[ 2x = 4 \] 3. Finally, divide by 2: \[ x = 2 \] So, the solution is \( x = 2 \). Thus, the answer is \( \text{D. } x = 2 \).
