Sure, let's solve the equation [tex]\(7^{x+1} = 7^{5-4x}\)[/tex] step by step.
1. Understand the equality of exponents:
Since the bases on both sides of the equation are the same (both 7), we can set the exponents equal to each other:
[tex]\[
x + 1 = 5 - 4x
\][/tex]
2. Combine like terms:
To simplify this, we need to get all [tex]\(x\)[/tex]-terms on one side of the equation. First, add [tex]\(4x\)[/tex] to both sides to combine the [tex]\(x\)[/tex]-terms:
[tex]\[
x + 4x + 1 = 5 - 4x + 4x
\][/tex]
This simplifies to:
[tex]\[
5x + 1 = 5
\][/tex]
3. Isolate the variable:
Next, subtract 1 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
5x + 1 - 1 = 5 - 1
\][/tex]
Simplifying further, we get:
[tex]\[
5x = 4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Finally, divide both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4}{5}
\][/tex]
So, the solution to the equation [tex]\(7^{x+1} = 7^{5-4x}\)[/tex] is:
[tex]\[
x = 0.8
\][/tex]
