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If [tex]7^{6x - 11} = 7^{2x + 1}[/tex], what is the value of [tex]x[/tex]?
To solve the equation [tex]\(7^{6x - 11} = 7^{2x + 1}\)[/tex], we can take advantage of the fact that the bases on both sides of the equation are the same. When the bases are identical, we can set the exponents equal to each other. So we start by equating the exponents:
[tex]\[6x - 11 = 2x + 1\][/tex]
Next, we solve the equation for [tex]\(x\)[/tex]. To do this, we'll follow these steps:
1. Subtract [tex]\(2x\)[/tex] from both sides to isolate the terms involving [tex]\(x\)[/tex] on one side:
[tex]\[
6x - 2x - 11 = 2x - 2x + 1
\][/tex]
This simplifies to:
[tex]\[
4x - 11 = 1
\][/tex]
2. Add 11 to both sides to further isolate the term with [tex]\(x\)[/tex]:
[tex]\[
4x - 11 + 11 = 1 + 11
\][/tex]
This simplifies to:
[tex]\[
4x = 12
\][/tex]
3. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{4x}{4} = \frac{12}{4}
\][/tex]
This simplifies to:
[tex]\[
x = 3
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is [tex]\(3\)[/tex].
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