Answer :
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]
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]
