Solve the following equations.Check your result in each case
`(y+7)/3=1+(3y-2)/5`

Answer

To solve the equation \(\frac{y + 7}{3} = 1 + \frac{3y - 2}{5}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{y + 7}{3} = 1 + \frac{3y - 2}{5} \]

### Step 2: Eliminate the fractions
To eliminate the fractions, we can find a common denominator. The least common multiple (LCM) of 3 and 5 is 15. We will multiply both sides of the equation by 15:
\[ 15 \cdot \frac{y + 7}{3} = 15 \cdot \left(1 + \frac{3y - 2}{5}\right) \]

### Step 3: Simplify both sides
Now, we simplify both sides:
\[ 5(y + 7) = 15 + 3(3y - 2) \]

### Step 4: Distribute on both sides
Distributing on both sides gives us:
\[ 5y + 35 = 15 + 9y - 6 \]

### Step 5: Combine like terms
Now, we combine like terms on the right side:
\[ 5y + 35 = 9y + 9 \]

### Step 6: Rearrange the equation
Next, we will move all terms involving \(y\) to one side and constant terms to the other side:
\[ 35 - 9 = 9y - 5y \]
This simplifies to:
\[ 26 = 4y \]

### Step 7: Solve for \(y\)
Now, we divide both sides by 4 to solve for \(y\):
\[ y = \frac{26}{4} = \frac{13}{2} \]

### Step 8: Verify the solution
To verify our solution, we substitute \(y = \frac{13}{2}\) back into the original equation:
\[ \frac{\frac{13}{2} + 7}{3} = 1 + \frac{3 \cdot \frac{13}{2} - 2}{5} \]

Calculating the left side:
\[ \frac{\frac{13}{2} + \frac{14}{2}}{3} = \frac{\frac{27}{2}}{3} = \frac{27}{6} = \frac{9}{2} \]

Calculating the right side:
\[ 1 + \frac{\frac{39}{2} - 2}{5} = 1 + \frac{\frac{39}{2} - \frac{4}{2}}{5} = 1 + \frac{\frac{35}{2}}{5} = 1 + \frac{35}{10} = 1 + \frac{7}{2} = \frac{2}{2} + \frac{7}{2} = \frac{9}{2} \]

Since both sides are equal, our solution is verified.

### Final Answer
\[ y = \frac{13}{2} \]