Solve : `(2y)/3-5/8=3/4-y/2`

Answer

To solve the equation \(\frac{2y}{3} - \frac{5}{8} = \frac{3}{4} - \frac{y}{2}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{2y}{3} - \frac{5}{8} = \frac{3}{4} - \frac{y}{2} \]

### Step 2: Move all terms involving \(y\) to one side and constant terms to the other side
Add \(\frac{y}{2}\) to both sides and add \(\frac{5}{8}\) to both sides:
\[ \frac{2y}{3} + \frac{y}{2} = \frac{3}{4} + \frac{5}{8} \]

### Step 3: Find a common denominator
For the left side, the denominators are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6. For the right side, the denominators are 4 and 8. The LCM of 4 and 8 is 8.

### Step 4: Rewrite the fractions with the common denominator
Convert the left side:
\[ \frac{2y \cdot 2}{3 \cdot 2} + \frac{y \cdot 3}{2 \cdot 3} = \frac{4y}{6} + \frac{3y}{6} = \frac{7y}{6} \]
Convert the right side:
\[ \frac{3 \cdot 2}{4 \cdot 2} + \frac{5}{8} = \frac{6}{8} + \frac{5}{8} = \frac{11}{8} \]

### Step 5: Set the two sides equal
Now we have:
\[ \frac{7y}{6} = \frac{11}{8} \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 7y \cdot 8 = 11 \cdot 6 \]
This simplifies to:
\[ 56y = 66 \]

### Step 7: Solve for \(y\)
Divide both sides by 56:
\[ y = \frac{66}{56} \]
This can be simplified:
\[ y = \frac{33}{28} \]

### Final Answer
Thus, the solution to the equation is:
\[ y = \frac{33}{28} \]

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