Question

solve for t.
$-\frac{4}{3}t + \frac{1}{2} = \frac{3}{2} - 2t + \frac{5}{3}t$
$t = $

Explanation:

Step1: Combine like terms on the right side

First, we combine the \( t \)-terms on the right side of the equation. The \( t \)-terms are \( -2t \) and \( \frac{5}{3}t \). To combine them, we need a common denominator. \( -2t \) can be written as \( -\frac{6}{3}t \). So, \( -\frac{6}{3}t + \frac{5}{3}t = -\frac{1}{3}t \). Now the equation becomes:
\[
-\frac{4}{3}t + \frac{1}{2} = \frac{3}{2} - \frac{1}{3}t
\]

Step2: Add \(\frac{4}{3}t\) to both sides

To get all the \( t \)-terms on one side, we add \( \frac{4}{3}t \) to both sides of the equation. This gives:
\[
\frac{1}{2} = \frac{3}{2} - \frac{1}{3}t + \frac{4}{3}t
\]
Simplifying the right side, \( -\frac{1}{3}t + \frac{4}{3}t = \frac{3}{3}t = t \). So now the equation is:
\[
\frac{1}{2} = \frac{3}{2} + t
\]

Step3: Subtract \(\frac{3}{2}\) from both sides

To solve for \( t \), we subtract \( \frac{3}{2} \) from both sides.
\[
\frac{1}{2} - \frac{3}{2} = t
\]
Simplifying the left side, \( \frac{1 - 3}{2} = \frac{-2}{2} = -1 \). So we have:
\[
t = -1
\]

Answer:

\( t = -1 \)