Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
