Solve the following equations :
`(a+5)/(6)-(a+1)/(9)=(a+3)/(4)`

Answer

To solve the equation \(\frac{a+5}{6} - \frac{a+1}{9} = \frac{a+3}{4}\), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the denominators 6, 9, and 4.

- The LCM of 6, 9, and 4 is 36.

### Step 2: Rewrite Each Fraction
Now, we will rewrite each fraction with the common denominator of 36:

- For \(\frac{a+5}{6}\):
\[ \frac{a+5}{6} = \frac{6(a+5)}{36} = \frac{6a + 30}{36} \]

- For \(\frac{a+1}{9}\):
\[ \frac{a+1}{9} = \frac{4(a+1)}{36} = \frac{4a + 4}{36} \]

- For \(\frac{a+3}{4}\):
\[ \frac{a+3}{4} = \frac{9(a+3)}{36} = \frac{9a + 27}{36} \]

### Step 3: Substitute Back into the Equation
Now, substitute these back into the equation:
\[ \frac{6a + 30}{36} - \frac{4a + 4}{36} = \frac{9a + 27}{36} \]

### Step 4: Combine the Left Side
Combine the fractions on the left side:
\[ \frac{(6a + 30) - (4a + 4)}{36} = \frac{9a + 27}{36} \]
This simplifies to:
\[ \frac{6a + 30 - 4a - 4}{36} = \frac{9a + 27}{36} \]
\[ \frac{2a + 26}{36} = \frac{9a + 27}{36} \]

### Step 5: Eliminate the Denominator
Since both sides have the same denominator, we can eliminate it:
\[ 2a + 26 = 9a + 27 \]

### Step 6: Rearranging the Equation
Now, rearranging the equation to isolate \(a\):
\[ 2a - 9a = 27 - 26 \]
\[ -7a = 1 \]

### Step 7: Solve for \(a\)
Now, divide both sides by -7:
\[ a = -\frac{1}{7} \]

### Final Answer
Thus, the solution to the equation is:
\[ a = -\frac{1}{7} \]

---