Solve for t.

Steps to Solve

1. Combine like terms on the right side

We start by combining the terms involving $t$ on the right side of the equation. The equation is given as: $$ \frac{-4}{3}t + \frac{1}{2} = \frac{3}{2} - 2t + \frac{5}{3}t $$

We can rewrite the right side: $$ -2t + \frac{5}{3}t = \left(-2 + \frac{5}{3}\right)t $$

Convert $-2$ to a fraction: $$ -2 = -\frac{6}{3} $$

Now combine the coefficients: $$ \left(-\frac{6}{3} + \frac{5}{3}\right)t = -\frac{1}{3}t $$

So the equation updates to: $$ \frac{-4}{3}t + \frac{1}{2} = \frac{3}{2} - \frac{1}{3}t $$

2. Isolate the terms involving t

Next, we want to isolate the $t$ terms on one side. Let's first add $\frac{1}{3}t$ to both sides: $$ \frac{-4}{3}t + \frac{1}{3}t + \frac{1}{2} = \frac{3}{2} $$

Simplifying the left side gives: $$ \frac{-4 + 1}{3}t + \frac{1}{2} = \frac{3}{2} $$

This simplifies to: $$ \frac{-3}{3}t + \frac{1}{2} = \frac{3}{2} $$ So it becomes: $$ -t + \frac{1}{2} = \frac{3}{2} $$

3. Solve for t

Now we need to isolate $t$. First, subtract $\frac{1}{2}$ from both sides: $$ -t = \frac{3}{2} - \frac{1}{2} $$

Calculate the right side: $$ -t = \frac{3 - 1}{2} = \frac{2}{2} = 1 $$

Now, multiply both sides by -1 to solve for $t$: $$ t = -1 $$