Answer :
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.
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.
