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, let’s simplify each expression one by one.

1. **For the first expression:**
\[
-\frac{15}{2}\left(\frac{4}{5}x - \frac{14}{15}\right)
\]
Distributing \(-\frac{15}{2}\):
\[
= -\frac{15}{2} \cdot \frac{4}{5}x + -\frac{15}{2} \cdot -\frac{14}{15}
\]
Simplifying each term:
- For the \(x\) term:
\[
-\frac{15}{2} \cdot \frac{4}{5} = -\frac{60}{10} = -6x
\]
- For the constant term:
\[
-\frac{15}{2} \cdot -\frac{14}{15} = \frac{14}{2} = 7
\]
Thus, the first expression simplifies to:
\[
-6x + 7
\]

2. **For the second expression:**
\[
\frac{5}{2}x + \frac{14}{3} + x + \frac{14}{4}
\]
First, convert \(x\) and \(\frac{14}{4}\) to have a common denominator:
\[
x = \frac{2}{2}x
\]
The common denominator for \(\frac{14}{3}\) and \(\frac{14}{4}\) is 12:
\[
\frac{14}{3} = \frac{56}{12}, \quad \frac{14}{4} = \frac{42}{12}
\]
Hence, the expression becomes:
\[
\frac{5}{2}x + \frac{56}{12} + \frac{2}{2}x + \frac{42}{12}
\]
Combine terms:
\[
= \left(\frac{5}{2} + 1\right)x + \left(\frac{56 + 42}{12}\right)
= \left(\frac{5}{2} + \frac{2}{2}\right)x + \frac{98}{12}
\]
Simplifying:
\[
= \left(\frac{7}{2}\right)x + \frac{49}{6}
= 3.5x + 8.1667 \text{ (approximately)}
\]

3. **For the third expression:**
\[
(-\frac{2}{3}+3)-(-4+\frac{16}{3})
\]
Simplifying inside the parentheses:
\[
= -\frac{2}{3} + 3 + 4 - \frac{16}{3}
\]
Convert \(3\) to thirds:
\[
3 = \frac{9}{3}
\]
Combine terms:
\[
= \left(-\frac{2}{3} + \frac{9}{3} - \frac{16}{3} + 4\right)
= \left(\frac{-2 + 9 - 16}{3} + 4\right)
= \left(-\frac{9}{3} + 4\right)
= -3 + 4 = 1
\]

Now we can compare the expressions:
1. The first expression simplifies to **\(-6x + 7\)**.
2. The second expression simplifies to approximately **\(3.5x + 8.1667\)**.
3. The third expression simplifies to **1**.

**Conclusion:**
None of the provided expressions are equivalent to each other based on the simplifications performed. If you meant to ask which two of these are equivalent, please provide further context or check if there are variations of the expressions that you intended to consider.