To determine which two expressions are equivalent, we can simplify each one.
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Expression 1: \(-\frac{15}{2}\left(\frac{4}{5}x - \frac{14}{15}\right)\)
Distributing \(-\frac{15}{2}\):
\[ -\frac{15}{2} \cdot \frac{4}{5}x + -\frac{15}{2} \cdot -\frac{14}{15} \]
This gives:
\[ -\frac{15 \cdot 4}{2 \cdot 5} x + \frac{15 \cdot 14}{2 \cdot 15} \] \[ = -\frac{60}{10} x + \frac{14}{2} \] \[ = -6x + 7 \]
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Expression 2: \(\frac{5}{2}x + \frac{14}{3} + x + \frac{14}{4}\)
First, we find a common denominator for \(\frac{14}{3}\) and \(\frac{14}{4}\):
\(\frac{14}{4} = \frac{14 \cdot 3.5}{4 \cdot 3.5} = \frac{42}{12}\) is equivalent to \(\frac{14}{4}\).
Then we need to find a common denominator for \(3\) and \(4\) which is \(12\).
Then we rewrite:
\[ = \frac{5}{2}x + \frac{56}{12} + \frac{12}{12} + \frac{42}{12} \] \[ = \frac{5}{2}x + \frac{56 + 12 + 42}{12} \] \[ = \frac{5}{2}x + \frac{110}{12} \] \[ = \frac{5}{2}x + \frac{55}{6} \]
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Expression 3: \((- \frac{2}{3} + 3) - (-4 + \frac{16}{3})\)
Starting with:
\[ -\frac{2}{3} + 3 + 4 - \frac{16}{3} \]
We will convert \(3\) and \(4\) into thirds:
\[ 3 = \frac{9}{3} \quad \text{and} \quad 4 = \frac{12}{3} \]
So we rewrite:
\[ = -\frac{2}{3} + \frac{9 + 12 - 16}{3} \] \[ = -\frac{2}{3} + \frac{5}{3} \] \[ = \frac{3}{3} \] \[ = 1 \]
Now, we compare the simplified results from the three expressions:
- From Expression 1: \(-6x + 7\)
- From Expression 2: \(\frac{5}{2}x + \frac{55}{6}\)
- From Expression 3: \(1\)
None of the expressions seem to be equivalent to each other upon simplification.
Thus, concluded: None of the three expressions provided are equivalent.