Solve : `(2x)/(3)-(x-1)/(6)+(7x-1)/(4)=2(1)/(6)`.
Hence, find the value of 'a', if `(1)/(a)+5x=8`.

Answer

To solve the equation \(\frac{2x}{3} - \frac{x-1}{6} + \frac{7x-1}{4} = 2 \frac{1}{6}\), we will follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \(2 \frac{1}{6}\) can be converted to an improper fraction:
\[ 2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6} \]
So the equation becomes:
\[ \frac{2x}{3} - \frac{x-1}{6} + \frac{7x-1}{4} = \frac{13}{6} \]

### Step 2: Find the LCM of the denominators
The denominators are \(3\), \(6\), and \(4\). The least common multiple (LCM) of these numbers is \(12\).

### Step 3: Multiply through by the LCM
Multiply every term in the equation by \(12\) to eliminate the fractions:
\[ 12 \left(\frac{2x}{3}\right) - 12 \left(\frac{x-1}{6}\right) + 12 \left(\frac{7x-1}{4}\right) = 12 \left(\frac{13}{6}\right) \]
This simplifies to:
\[ 4 \cdot 2x - 2 \cdot (x - 1) + 3 \cdot (7x - 1) = 26 \]
Which further simplifies to:
\[ 8x - 2(x - 1) + 21x - 3 = 26 \]

### Step 4: Distribute and combine like terms
Distributing the \(-2\) gives:
\[ 8x - 2x + 2 + 21x - 3 = 26 \]
Combine like terms:
\[ (8x - 2x + 21x) + (2 - 3) = 26 \]
This simplifies to:
\[ 27x - 1 = 26 \]

### Step 5: Solve for \(x\)
Add \(1\) to both sides:
\[ 27x = 27 \]
Now, divide by \(27\):
\[ x = 1 \]

### Step 6: Substitute \(x\) into the second equation
Now we need to find the value of \(a\) using the equation \(\frac{1}{a} + 5x = 8\):
Substituting \(x = 1\):
\[ \frac{1}{a} + 5(1) = 8 \]
This simplifies to:
\[ \frac{1}{a} + 5 = 8 \]

### Step 7: Solve for \(a\)
Subtract \(5\) from both sides:
\[ \frac{1}{a} = 3 \]
Now, taking the reciprocal gives:
\[ a = \frac{1}{3} \]

### Final Answer
Thus, the values are:
\[ x = 1 \quad \text{and} \quad a = \frac{1}{3} \]