Let's analyze Noah's work on both equations step-by-step and identify where and why he ended up with false statements.
### Equation 1:
[tex]\[
x + 6 = 4x + 1 - 3x
\][/tex]
#### Noah's Steps:
1. Combine like terms:
[tex]\[
x + 6 = 4x - 3x + 1
\][/tex]
Which simplifies to:
[tex]\[
x + 6 = x + 1
\][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
x + 6 - x = x + 1 - x
\][/tex]
Simplifies to:
[tex]\[
6 = 1
\][/tex]
At this stage, Noah ends up with the false statement [tex]\(6 = 1\)[/tex].
#### Observations:
- Noah's moves were technically acceptable; however, the original equation [tex]\(x + 6 = 4x + 1 - 3x\)[/tex] simplifies to an identity [tex]\(x + 6 = x + 1\)[/tex].
- By subtracting [tex]\(x\)[/tex] from both sides, we get [tex]\(6 = 1\)[/tex], which is a contradiction.
#### Conclusion:
The contradiction here indicates that there is no solution to this equation. The terms and steps are algebraically correct, but the equation itself is impossible to solve because it results in a false statement.
### Equation 2:
[tex]\[
2(5 + x) - 1 = 3x + 9
\][/tex]
#### Noah's Steps:
1. Apply the distributive property:
[tex]\[
2 \cdot 5 + 2 \cdot x - 1 = 3x + 9
\][/tex]
Simplifies to:
[tex]\[
10 + 2x - 1 = 3x + 9
\][/tex]
Which further simplifies to:
[tex]\[
9 + 2x = 3x + 9
\][/tex]
2. Subtract 10 from each side (This step is incorrect as written; correctly, we should start by subtracting 9):
[tex]\[
9 + 2x - 9 = 3x + 9 - 9
\][/tex]
Simplifies to:
[tex]\[
2x = 3x
\][/tex]
3. Subtract 2x from each side:
[tex]\[
2x - 2x = 3x - 2x
\][/tex]
Simplifies to:
[tex]\[
0 = x
\][/tex]
Noah's final step divider each sides by x leads him to false statement [tex]\(2 = 3\)[/tex], which is incorrect.
#### Observation:
- Noah should not divide each side by [tex]\(x\)[/tex] after subtracting [tex]\(2x\)[/tex] otherwise he would factor out [tex]\(x\)[/tex] and obtain correct solution without breaking the equation.
- Correctly executing the steps, we see:
[tex]\(\text{Subtract } 2x \text{ from each side:}\)[/tex]
[tex]\[
9 + 2x - 2 \rightarrow x = 0 \\
_0 + 9 \rightarrow remain true statement
\][/tex]
#### Conclusion:
The equation simplifies correctly when solving step-by-step without dividing either side by [tex]\(x\)[/tex] after simplifying subtraction to yield a correct valid solution:
[tex]\[ x = 0 \][/tex]
Noah's moves would have been correct without the erroneous step and correct answer would be identified earlier.
