Let's solve the given equation step-by-step to find and correct any mistakes.
### Original Equation:
[tex]\[ 6x - 1 = -2x + 9 \][/tex]
### Step-by-Step Solution:
1. Combine like terms on both sides of the equation:
Add [tex]\(2x\)[/tex] to both sides to get all terms with [tex]\(x\)[/tex] on the left side:
[tex]\[
6x - 1 + 2x = -2x + 9 + 2x
\][/tex]
Simplifying, we get:
[tex]\[
8x - 1 = 9
\][/tex]
2. Isolate the term with [tex]\(x\)[/tex]:
Add 1 to both sides to move the constant term to the right side:
[tex]\[
8x - 1 + 1 = 9 + 1
\][/tex]
Simplifying, we get:
[tex]\[
8x = 10
\][/tex]
3. Solve for [tex]\(x\)[/tex] by dividing both sides by 8:
[tex]\[
x = \frac{10}{8}
\][/tex]
4. Simplify the fraction:
[tex]\[
x = \frac{10}{8} = \frac{5}{4}
\][/tex]
### Identifying the Mistake:
1. Given Steps:
[tex]\[
\begin{aligned}
6x - 1 & = -2x + 9 \\
8x - 1 & = 9 \\
8x & = 10 \\
x & = \frac{8}{10} \\
x & = \frac{4}{5}
\end{aligned}
\][/tex]
2. Step 3:
The step where we solve [tex]\(8x = 10\)[/tex] by dividing both sides by 8 should give:
[tex]\[
x = \frac{10}{8}
\][/tex]
This step is correctly performed (although the simplification afterward is incorrect).
3. Step 4:
The simplification of [tex]\(x = \frac{10}{8}\)[/tex] was incorrectly shown as [tex]\( \frac{4}{5} \)[/tex]. The correct simplification is:
[tex]\[
x = \frac{5}{4}
\][/tex]
Thus, the correct answer is:
C. Step 3 is incorrect and should be [tex]\( x = \frac{10}{8} \)[/tex].
To summarize: The mistake in the solution is in the simplification of the fraction in Step 3. It should be [tex]\( x = \frac{10}{8} \)[/tex] and simplify to [tex]\( \frac{5}{4} \)[/tex], not [tex]\( \frac{4}{5} \)[/tex].
