To match the pairs of equivalent expressions, we need to simplify each of the given combinations step by step:
1. Simplify the expression:
[tex]\[
\left(4t - \frac{8}{5}\right) - \left(3 - \frac{4}{3} t\right)
\][/tex]
First, distribute the negative sign inside the parenthesis:
[tex]\[
4t - \frac{8}{5} - 3 + \frac{4}{3} t
\][/tex]
Combine like terms:
[tex]\[
4t + \frac{4}{3}t - \frac{8}{5} - 3
\][/tex]
To combine \(4t + \frac{4}{3}t\), find a common denominator:
[tex]\[
4t = \frac{12}{3}t \implies \frac{12}{3}t + \frac{4}{3}t = \frac{16}{3}t
\][/tex]
So, the expression simplifies to:
[tex]\[
\frac{16}{3}t - \frac{8}{5} - 3
\][/tex]
Convert -3 to a fraction with the same denominator as \(\frac{8}{5}\):
[tex]\[
-3 = -\frac{15}{5}
\][/tex]
Combine \(\frac{8}{5} + \frac{15}{5}\):
[tex]\[
\frac{16}{3}t - \left(\frac{8}{5} + \frac{15}{5}\right) = \frac{16}{3}t - \frac{23}{5}
\][/tex]
So, \(\left(4 t-\frac{8}{5}\right)-\left(3-\frac{4}{3} t\right)\) matches \(\frac{16}{3} t-\frac{23}{5}\).
2. Simplify the expression:
[tex]\[
3(3t - 4) - (2t + 10)
\][/tex]
First, distribute the 3:
[tex]\[
9t - 12 - 2t - 10
\][/tex]
Combine like terms:
[tex]\[
(9t - 2t) - (12 + 10) = 7t - 22
\][/tex]
So, \(3(3 t-4)-(2 t+10)\) matches \(7 t-22\).
3. Simplify the expression:
[tex]\[
5(2t + 1) + (-7t + 28)
\][/tex]
First, distribute the 5:
[tex]\[
10t + 5 + (-7t + 28)
\][/tex]
Combine like terms:
[tex]\[
(10t - 7t) + (5 + 28) = 3t + 33
\][/tex]
So, \(5(2 t+1)+(-7 t+28)\) matches \(3 t+33\).
4. Simplify the expression:
[tex]\[
\left(-\frac{9}{2} t+3\right) + \left(\frac{7}{4} t+33\right)
\][/tex]
First, find a common denominator for the terms involving \(t\):
[tex]\[
-\frac{9}{2} t = -\frac{18}{4} t \implies -\frac{18}{4} t + \frac{7}{4} t = -\frac{11}{4} t
\][/tex]
Combine constant terms:
[tex]\[
3 + 33 = 36
\][/tex]
So, \(\left(-\frac{9}{2} t+3\right)+\left(\frac{7}{4} t+33\right)\) matches \(-\frac{11}{4} t+36\).
Thus, the pairs of equivalent expressions are:
[tex]\[
\begin{array}{ccc}
\left(4 t-\frac{8}{5}\right)-\left(3-\frac{4}{3} t\right) & \text{matches} & \frac{16}{3} t-\frac{23}{5} \\
3(3 t-4)-(2 t+10) & \text{matches} & 7 t-22 \\
5(2 t+1)+(-7 t+28) & \text{matches} & 3 t+33 \\
\left(-\frac{9}{2} t+3\right)+\left(\frac{7}{4} t+33\right) & \text{matches} & -\frac{11}{4} t+36 \\
\end{array}
\][/tex]
