Select the correct answer.

Find the mistake made in the steps to solve the equation below.

[tex]\[
\begin{aligned}
6x - 1 & = -2x + 9 \\
8x - 1 & = 9 \\
8x & = 10 \\
x & = \frac{8}{10} \\
x & = \frac{4}{5}
\end{aligned}
\][/tex]

1. Addition property of equality
2. Addition property of equality
3. Division property of equality
4. Simplification

A. The justification for step 2 is incorrect and should be the subtraction property of equality.
B. The justification for step 3 is incorrect and should be the multiplication property of equality.
C. Step 3 is incorrect and should be [tex]\[ x = \frac{10}{8} \][/tex].
D. Step 2 is incorrect and should be [tex]\[ 8x = 8 \][/tex].



Answer :

To identify the mistake in solving the equation [tex]\(6x - 1 = -2x + 9\)[/tex], let's go through each step methodically and check for correctness and proper justifications:

1. Starting equation:
[tex]\(6x - 1 = -2x + 9\)[/tex]

2. Combining like terms:
Adding [tex]\(2x\)[/tex] to both sides of the equation to combine like terms on one side.
[tex]\[ 6x - 1 + 2x = -2x + 9 + 2x \][/tex]
Simplifies to:
[tex]\[ 8x - 1 = 9 \][/tex]
The justification for this step is the Addition property of equality, which is correctly applied.

3. Removing the constant term from one side:
Adding 1 to both sides to isolate the term involving `x`.
[tex]\[ 8x - 1 + 1 = 9 + 1 \][/tex]
Simplifies to:
[tex]\[ 8x = 10 \][/tex]
The justification for this step is the Addition property of equality, which is correctly applied.

4. Solving for [tex]\(x\)[/tex]:
Dividing both sides by 8 to solve for `x`.
[tex]\[ \frac{8x}{8} = \frac{10}{8} \][/tex]
Simplifies to:
[tex]\[ x = \frac{10}{8} \][/tex]
The justification for this step is the Division property of equality, which is correctly applied.

5. Simplifying the fraction:
Simplifying the fraction:
[tex]\[ x = \frac{10}{8} = \frac{5}{4} \][/tex]

Given the process, we notice an inconsistency in the provided solution steps. In step 4, the simplification results in:
[tex]\[ x = \frac{10}{8} \][/tex]
However, there's a mistake in stating the intermediate result directly as:
[tex]\[ x = \frac{8}{10} \][/tex]
This is incorrect since it should be [tex]\(x = \frac{10}{8}\)[/tex].

Therefore, the correct answer that identifies the mistake is:

C. Step 3 is incorrect and should be [tex]\( x = \frac{10}{8} \)[/tex].