Sure, let's thoroughly analyze and match each expression to their equivalent simplified form by comparing and simplifying each term individually.
1. Expression 1: \((4t - \frac{8}{5}) - (3 - \frac{4}{3} t)\)
[tex]\[
\text{Simplify the expression:} \\
(4t - \frac{8}{5}) - (3 - \frac{4}{3} t) \ = \ 4t - \frac{8}{5} - 3 + \frac{4}{3} t \\
\text{Combine like terms:} \\
\left(4 + \frac{4}{3} \right)t - 3 - \frac{8}{5} \\
= \left(\frac{12}{3} + \frac{4}{3}\right)t - \left(\frac{15}{5} + \frac{8}{5}\right) \\
= \left(\frac{16}{3} \right)t - \left(\frac{23}{5}\right) \\
\Rightarrow \frac{16}{3} t - \frac{23}{5}
\][/tex]
This expression does not seem to match with the result we verified, so we simplify it further:
Result:
[tex]\[
\text{Simplified form: } 5.33333333333333t - 4.6
\][/tex]
2. Expression 2: \(5(2t + 1) + (-7t + 28)\)
[tex]\[
\text{Expand and simplify the expression:} \\
5(2t + 1) + (-7t + 28) \\
= 10t + 5 - 7t + 28 \\
= 10t - 7t + 5 + 28 \\
= 3t + 33
\][/tex]
Result:
[tex]\[
\text{Simplified form: } 3t + 33
\][/tex]
3. Expression 3: \(\left(-\frac{9}{2}t + 3\right) + \left(\frac{7}{4}t + 33\right)\)
[tex]\[
\text{Combine like terms:} \\
-\frac{9}{2}t + \frac{7}{4}t + 3 + 33 \\
= -\left(\frac{18}{4}\right)t + \left(\frac{7}{4}\right)t + 36 \\
= -\left(\frac{18 - 7}{4}\right)t + 36 \\
= -\left(\frac{11}{4}\right)t + 36
\][/tex]
Result:
[tex]\[
\text{Simplified form: } -2.75t + 36
\][/tex]
4. Expression 4: \(3(3t - 4) - (2t + 10)\)
[tex]\[
\text{Expand the expression:} \\
3(3t - 4) - (2t + 10) \\
= 9t - 12 - 2t - 10 \\
= 9t - 2t - 12 - 10 \\
= 7t - 22
\][/tex]
Result:
[tex]\[
\text{Simplified form: } 7t - 22
\][/tex]
Now, let's match these results with the original given expressions.
- From our simplifications, we obtain:
- \(\frac{16}{3} t - \frac{23}{5}\) simplifies to \(5.33333333333333t - 4.6\)
- \(5(2t + 1) + (-7t + 28)\) simplifies to \(3t + 33\)
- \(\left(-\frac{9}{2}t + 3\right) + \left(\frac{7}{4}t + 33\right)\) simplifies to \(\frac{-11}{4}t + 36\)
- \(3(3t - 4) - (2t + 10)\) simplifies to \(7t - 22\)
The equivalent expressions are thus:
[tex]\[
\begin{array}{l}
(4t - \frac{8}{5}) - (3 - \frac{4}{3} t) \quad \equiv \quad 5.33333333333333t - 4.6 \\
7t - 22 \quad \equiv \quad 7t - 22 \\
5(2t + 1) + (-7t + 28) \quad \equiv \quad 3t + 33 \\
\frac{16}{3} t - \frac{23}{5} \quad \not\equiv \quad \text{(No matching result)} \\
\left(-\frac{9}{2} t + 3\right) + \left(\frac{7}{4} t + 33\right) \quad \equiv \quad \frac{-11}{4}t + 36 \\
-\frac{11}{4}t + 36 \quad \equiv \quad \frac{-11}{4}t + 36 \\
3(3t - 4) - (2t + 10) \quad \equiv \quad 7t - 22 \\
3t + 33 \quad \equiv \quad 3t + 33 \\
\end{array}
\][/tex]
These are the corresponding simplified expressions.
