Solve : `(2x+1)/(10)-(3-2x)/(15)=(x-2)/(6)`. Hence, find the value of y, if `(1)/(x)+(1)/(y)=3`.

Answer

To solve the equation \(\frac{2x+1}{10} - \frac{3-2x}{15} = \frac{x-2}{6}\), we will follow these steps:

### Step 1: Find the LCM
The denominators are 10, 15, and 6. The least common multiple (LCM) of these numbers is 30.

### Step 2: Rewrite the equation with the LCM
Multiply each term by 30 to eliminate the denominators:

\[ 30 \left(\frac{2x+1}{10}\right) - 30 \left(\frac{3-2x}{15}\right) = 30 \left(\frac{x-2}{6}\right) \]

This simplifies to:

\[ 3(2x + 1) - 2(3 - 2x) = 5(x - 2) \]

### Step 3: Distribute the terms
Distributing the terms gives us:

\[ 6x + 3 - 6 + 4x = 5x - 10 \]

This simplifies to:

\[ 10x - 3 = 5x - 10 \]

### Step 4: Move variables to one side and constants to the other
Subtract \(5x\) from both sides and add 3 to both sides:

\[ 10x - 5x = -10 + 3 \]

This simplifies to:

\[ 5x = -7 \]

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

\[ x = -\frac{7}{5} \]

### Step 6: Find the value of \(y\)
Now we need to find \(y\) using the equation \(\frac{1}{x} + \frac{1}{y} = 3\). Substitute \(x = -\frac{7}{5}\):

\[ \frac{1}{-\frac{7}{5}} + \frac{1}{y} = 3 \]

This simplifies to:

\[ -\frac{5}{7} + \frac{1}{y} = 3 \]

### Step 7: Isolate \(\frac{1}{y}\)
Add \(\frac{5}{7}\) to both sides:

\[ \frac{1}{y} = 3 + \frac{5}{7} \]

### Step 8: Find a common denominator
The common denominator for 3 (which is \(\frac{21}{7}\)) and \(\frac{5}{7}\) is 7:

\[ \frac{1}{y} = \frac{21}{7} + \frac{5}{7} = \frac{26}{7} \]

### Step 9: Solve for \(y\)
Taking the reciprocal gives:

\[ y = \frac{7}{26} \]

### Final Answer
The values are:
- \(x = -\frac{7}{5}\)
- \(y = \frac{7}{26}\)

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