To solve the equation [tex]\(\frac{5}{4} y - \frac{1}{3} = \frac{1}{2}(3y - 1)\)[/tex], let's go through the steps in detail:
1. Start with the given equation:
[tex]\[
\frac{5}{4}y - \frac{1}{3} = \frac{1}{2}(3y - 1)
\][/tex]
2. Distribute [tex]\(\frac{1}{2}\)[/tex] on the right-hand side:
[tex]\[
\frac{5}{4}y - \frac{1}{3} = \frac{1}{2} \cdot 3y - \frac{1}{2} \cdot 1
\][/tex]
[tex]\[
\frac{5}{4}y - \frac{1}{3} = \frac{3}{2}y - \frac{1}{2}
\][/tex]
3. To eliminate the fractions, find a common denominator. The least common multiple of 4, 3, and 2 is 12. Multiply every term by 12 to clear the denominators:
[tex]\[
12 \left(\frac{5}{4}y\right) - 12 \left(\frac{1}{3}\right) = 12 \left(\frac{3}{2}y\right) - 12 \left(\frac{1}{2}\right)
\][/tex]
[tex]\[
3 \cdot 5y - 4 = 6 \cdot 3y - 6
\][/tex]
[tex]\[
15y - 4 = 18y - 6
\][/tex]
4. Now, get all the [tex]\(y\)[/tex] terms on one side and the constant terms on the other side by subtracting [tex]\(15y\)[/tex] from both sides:
[tex]\[
15y - 4 - 15y = 18y - 6 - 15y
\][/tex]
[tex]\[
-4 = 3y - 6
\][/tex]
5. Add 6 to both sides to isolate the [tex]\(3y\)[/tex] term:
[tex]\[
-4 + 6 = 3y
\][/tex]
[tex]\[
2 = 3y
\][/tex]
6. Finally, solve for [tex]\(y\)[/tex] by dividing both sides by 3:
[tex]\[
y = \frac{2}{3}
\][/tex]
So the solution to the equation [tex]\(\frac{5}{4}y - \frac{1}{3} = \frac{1}{2}(3y - 1)\)[/tex] is [tex]\(y = \frac{2}{3}\)[/tex].
Comparing this with the given options:
A. [tex]\(y = -10\)[/tex]
B. [tex]\(y = \frac{68}{201}\)[/tex]
C. [tex]\(y = \frac{2}{3}\)[/tex]
D. [tex]\(y = 0\)[/tex]
The correct answer is:
C. [tex]\(y = \frac{2}{3}\)[/tex]
