Question

Which two are equivalent
-15/2(4/5x-14/15)
5/2x+14/3+x+14/4
(-2/3+3)-(-4+16/3)

Answer 1

To determine which two expressions are equivalent, we can simplify each one.

1. **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:

\[
-\frac{15 \cdot 4}{2 \cdot 5} x + \frac{15 \cdot 14}{2 \cdot 15}
\]
\[
= -\frac{60}{10} x + \frac{14}{2}
\]
\[
= -6x + 7
\]

2. **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:

\[
= \frac{5}{2}x + \frac{56}{12} + \frac{12}{12} + \frac{42}{12}
\]
\[
= \frac{5}{2}x + \frac{56 + 12 + 42}{12}
\]
\[
= \frac{5}{2}x + \frac{110}{12}
\]
\[
= \frac{5}{2}x + \frac{55}{6}
\]

3. **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:

1. From Expression 1: \(-6x + 7\)
2. From Expression 2: \(\frac{5}{2}x + \frac{55}{6}\)
3. 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.