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.
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
5 answers
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.
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.
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.
6 answers