Sure, let's solve the equation step by step:
[tex]\[
\frac{3t - 2}{4} - \frac{2t + 3}{3} = \frac{2}{3} - t
\][/tex]
1. Eliminate the fractions by finding a common denominator, which in this case is 12 (the least common multiple of 4 and 3). Multiply every term by 12 to clear the fractions:
[tex]\[
12 \left( \frac{3t - 2}{4} \right) - 12 \left( \frac{2t + 3}{3} \right) = 12 \left( \frac{2}{3} \right) - 12t
\][/tex]
2. Simplify each term:
[tex]\[
3t \cdot 3 - 2 \cdot 3 - 4 \cdot (2t + 3) = 4 \cdot 2 - 12t
\][/tex]
[tex]\[
9t - 6 - (8t + 12) = 8 - 12t
\][/tex]
3. Distribute and Combine like terms:
[tex]\[
9t - 6 - 8t - 12 = 8 - 12t
\][/tex]
[tex]\[
t - 18 = 8 - 12t
\][/tex]
4. Get all [tex]\( t \)[/tex]-terms on one side and the constants on the other side:
Add [tex]\( 12t \)[/tex] to both sides of the equation:
[tex]\[
t + 12t - 18 = 8
\][/tex]
Combine the [tex]\( t \)[/tex]-terms on the left-hand side:
[tex]\[
13t - 18 = 8
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Add 18 to both sides:
[tex]\[
13t = 26
\][/tex]
Divide both sides by 13:
[tex]\[
t = 2
\][/tex]
Thus, the solution to the equation is:
[tex]\[
t = 2
\][/tex]
