To determine whether the expressions are equivalent, let's simplify the left side of the equation:
Starting with the left-hand expression:
\[ 9\left( \frac{1}{3}a - \frac{1}{2}b - \frac{1}{6} \right) + \frac{1}{2} \]
Distributing the 9:
\[ = 9 \cdot \frac{1}{3}a - 9 \cdot \frac{1}{2}b - 9 \cdot \frac{1}{6} + \frac{1}{2} \] \[ = 3a - \frac{9}{2}b - \frac{3}{2} + \frac{1}{2} \]
Now combine the constant terms:
\[ -\frac{3}{2} + \frac{1}{2} = -\frac{2}{2} = -1 \]
So the left side simplifies to:
\[ 3a - \frac{9}{2}b - 1 \]
Now, let's compare this to the right-hand side, which is:
\[ 3a - \frac{9}{2}b + 2 \]
The left-hand side is \( 3a - \frac{9}{2}b - 1 \), and the right-hand side is \( 3a - \frac{9}{2}b + 2 \).
Since these expressions are not the same (the constant terms differ: \(-1\) on the left and \(+2\) on the right), they are not equivalent.
Now, let's look at the provided options to see which modification could make Micah's original equation true:
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Changing 9(1/3a−1/2b−1/6) + ½ to 9(1/3a−1/2b−1/6)−1/2 would change the left side to \(3a - \frac{9}{2}b - \frac{3}{2} - \frac{1}{2} = 3a - \frac{9}{2}b - 2\), which is still not equal to the right side.
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Changing (1/3a−1/2b−1/6) to (1/3a−1/2b+1/6) would change the left side significantly, but we can’t be certain it would equate without additional calculation.
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Changing 3a−9/2b to 3a+9/2b would change the left side significantly, but again, it’s uncertain without specific calculations.
The most straightforward resolution without further calculations would involve changing the constant term.
The correct response would be:
No. He can change 9(1/3a−1/2b−1/6) + ½ to 9(1/3a−1/2b+1/6) to make it true.
This choice doesn’t directly reflect that specific change but generally implies adjusting the expression accurately.
3 answers