Noah is having trouble solving two equations. In each case he took steps that he thought were acceptable but ended up with statements that are clearly not true. Analyze Noah's work on each equation and the moves he made. Were they acceptable moves? Why do you think he ended up with a false equation.
1. x + 6 = 4x + 1 - 3x
x + 6 = 4x - 3x + 1 (Apply the commutative property)
x + 6 = x + 1 Combine like terms
6 = 1Subtract x from each side
2. 2(5 + x) - 1 = 3x + 9
10 + 2x - 1 = 3x + 9 apply the distributive property
2x - 1 = 3x - 1 subtract 10 from each side
2x = 3x add 1 to each side
2 = 3 divide each side by x
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.
What is the answer to both?
1. Let's correct Noah's mistakes in solving the first equation:
x + 6 = 4x + 1 - 3x
x + 6 = x + 1 (Combine the like terms correctly)
6 = 1 (Subtract x from each side)
We can see that Noah's mistake was in erroneously combining like terms on the left side of the equation. It should have been x + 6 = x + 1, not x + 6 = 1.
Therefore, the first equation has no solution as the simplified equation 6 = 1 is not true.
2. Now, let's correct Noah's mistakes in solving the second equation:
2(5 + x) - 1 = 3x + 9
10 + 2x - 1 = 3x + 9 (Apply the distributive property correctly)
2x - 1 = 3x + 8 (Combine the like terms correctly)
-1 = x + 8 (Subtract 3x from both sides)
Noah's mistakes were in incorrectly subtracting 1 from the left side of the equation and mistakenly adding 1 to both sides instead of adding it specifically to the x term on the right side.
Therefore, the corrected equation is -1 = x + 8.
I'm a bit confused on number 2. Shouldn't x = 0?
Apologies for the confusion. Let's correct the mistake in the second equation once more:
2(5 + x) - 1 = 3x + 9
10 + 2x - 1 = 3x + 9 (Apply the distributive property correctly)
9 + 2x = 3x + 9 (Combine the like terms correctly)
2x - 3x = 9 - 9 (Subtract 9 from both sides)
-x = 0 (Combine the like terms correctly)
x = 0 (Multiply both sides by -1 to isolate x)
You are correct, the correct solution to the equation 2(5 + x) - 1 = 3x + 9 is x = 0.
Let's analyze Noah's work for each equation separately:
1. In the first equation, Noah applied the commutative property when he rearranged the terms. This is an acceptable move since the commutative property allows the order of addition or multiplication to be changed without affecting the result.
However, the mistake occurred when Noah combined like terms. He should have obtained 3x on the right side instead of x. This mistake led to the false equation 6 = 1, which is clearly not true.
2. In the second equation, Noah correctly applied the distributive property to simplify the left side of the equation. This is an acceptable move.
However, the error occurred when Noah subtracted 10 from both sides of the equation. He should have subtracted 2x instead. This mistake led to the false equation 2x - 1 = 3x - 1.
Additionally, when Noah added 1 to both sides, he incorrectly simplified the left side. The correct result would have been 2x + 9 on the left side.
Finally, when Noah divided each side by x, this move is not acceptable since we do not know if x is equal to zero or not. Dividing by x can introduce extraneous solutions or lead to division by zero, both of which can result in false equations.
To summarize, the moves made by Noah in each equation were not all acceptable. The mistakes he made resulted in false equations.
Let's analyze Noah's work on each equation and determine whether his moves were acceptable and why he ended up with a false equation.
1. x + 6 = 4x + 1 - 3x
Noah started by applying the commutative property to rearrange the terms. This step is acceptable. However, Noah made a mistake when combining like terms. The correct step would be to simplify both sides of the equation:
x + 6 = x + 1
Subtracting x from each side, we get:
6 = 1
Noah ended up with a false equation because at the step of combining like terms, he incorrectly canceled out the x terms on both sides. This led to an inconsistency where 6 is equal to 1, which is not possible.
2. 2(5 + x) - 1 = 3x + 9
Noah started by applying the distributive property, which is an acceptable move:
10 + 2x - 1 = 3x + 9
Next, Noah subtracted 10 from each side, which is also acceptable:
2x - 1 = 3x - 1
However, Noah made a mistake in the next step when he added 1 to each side. The correct step would be to add 1 to the right side, not both sides:
2x - 1 + 1 = 3x - 1 + 1
This simplifies to:
2x = 3x
Noah ended up with a false equation because he incorrectly canceled out the constant term -1 on both sides. This led to a contradiction where 2x is equal to 3x, which is not possible.
In both cases, Noah made mistakes when combining like terms or canceling out terms that should not have been canceled. These errors led to false equations and contradictions.