Solve for [tex]\( t \)[/tex].

[tex]\[
\frac{3t - 2}{4} - \frac{2t + 3}{3} = \frac{2}{3} - t
\][/tex]

Given [tex]\( t = 2 \)[/tex], check the solution.



Answer :

Certainly! Let's solve the equation:

[tex]\[ \frac{3t - 2}{4} - \frac{2t + 3}{3} = \frac{2}{3} - t \][/tex]

We need to check if [tex]\( t = 2 \)[/tex] is a solution to this equation.

Step 1: Substitute [tex]\( t = 2 \)[/tex] into the equation.

Substitute [tex]\( t = 2 \)[/tex] in the left-hand side (LHS) of the equation:

[tex]\[ \text{LHS} = \frac{3(2) - 2}{4} - \frac{2(2) + 3}{3} \][/tex]

Calculate each part:

[tex]\[ 3(2) - 2 = 6 - 2 = 4 \][/tex]

[tex]\[ \frac{4}{4} = 1 \][/tex]

Then,

[tex]\[ 2(2) + 3 = 4 + 3 = 7 \][/tex]

[tex]\[ \frac{7}{3} \][/tex]

So,

[tex]\[ \text{LHS} = 1 - \frac{7}{3} \][/tex]

Convert 1 to a fraction with a denominator of 3:

[tex]\[ 1 = \frac{3}{3} \][/tex]

Thus,

[tex]\[ \text{LHS} = \frac{3}{3} - \frac{7}{3} = \frac{3 - 7}{3} = \frac{-4}{3} \][/tex]

Step 2: Substitute [tex]\( t = 2 \)[/tex] into the right-hand side (RHS) of the equation.

Substitute [tex]\( t = 2 \)[/tex] in the RHS:

[tex]\[ \text{RHS} = \frac{2}{3} - 2 \][/tex]

Convert 2 to a fraction with a denominator of 3:

[tex]\[ 2 = \frac{6}{3} \][/tex]

So,

[tex]\[ \text{RHS} = \frac{2}{3} - \frac{6}{3} = \frac{2 - 6}{3} = \frac{-4}{3} \][/tex]

Step 3: Compare the LHS and RHS.

We have:

[tex]\[ \text{LHS} = \frac{-4}{3} \][/tex]

[tex]\[ \text{RHS} = \frac{-4}{3} \][/tex]

Since the LHS is equal to the RHS, the equation is satisfied.

Therefore, [tex]\( t = 2 \)[/tex] is indeed a solution to the equation

[tex]\[ \frac{3t - 2}{4} - \frac{2t + 3}{3} = \frac{2}{3} - t \][/tex]