Solve the following equations :
`(4(y+2))/(5)=7+(5y)/(13)`

Answer

To solve the equation \( \frac{4(y+2)}{5} = 7 + \frac{5y}{13} \), we will follow these steps:

### Step 1: Eliminate the fractions
Multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 65.

\[ 65 \cdot \frac{4(y+2)}{5} = 65 \cdot \left(7 + \frac{5y}{13}\right) \]

### Step 2: Simplify both sides
Distributing on both sides:

\[ 13 \cdot 4(y+2) = 65 \cdot 7 + 5 \cdot 5y \]

This simplifies to:

\[ 52(y + 2) = 455 + 25y \]

### Step 3: Expand the left side
Distributing \(52\) on the left side:

\[ 52y + 104 = 455 + 25y \]

### Step 4: Move all \(y\) terms to one side and constant terms to the other side
Subtract \(25y\) from both sides and subtract \(104\) from both sides:

\[ 52y - 25y = 455 - 104 \]

This simplifies to:

\[ 27y = 351 \]

### Step 5: Solve for \(y\)
Divide both sides by \(27\):

\[ y = \frac{351}{27} \]

### Step 6: Simplify the fraction
Calculating \( \frac{351}{27} \):

\[ y = 13 \]

Thus, the solution to the equation is:

\[ \boxed{13} \]

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