1. Noah's first mistake was applying the commutative property incorrectly. The commutative property states that the order of addition or multiplication does not affect the result. However, in this case, Noah mistakenly rearranged the terms on one side of the equation without changing their order of operations. This resulted in a false equation because the terms were not correctly simplified.
Noah's second mistake was combining like terms incorrectly. On the right side of the equation, Noah correctly combined 4x and -3x to get x. However, he incorrectly combined x and 1 on the left side of the equation, resulting in x + 1 instead of x + 6.
Lastly, Noah incorrectly subtracted x from both sides of the equation. This step is unnecessary and incorrect, as it changes the equation and does not help in simplifying it further.
2. Noah's first mistake in the second equation was also in applying the distributive property incorrectly. He correctly distributed the 2 to both terms inside the parentheses, but then incorrectly subtracted 1 instead of adding it.
Noah's second mistake was incorrect subtraction when he tried to isolate the variable. He subtracted 10 from both sides correctly, but then subtracted 1 from the left side of the equation instead of the right side.
Lastly, Noah incorrectly added 1 to both sides of the equation instead of adding it specifically to the x term on the right side. This led to the equation 2 = 3, which is clearly not true.
In both cases, Noah's mistakes were due to incorrect application of mathematical properties and operations, leading to false equations.