Solve: `(11)/(a +b) + 2 (a -b) = 11, (22)/(a +b) + 3(a -b) = 17`

Answer

To solve the equations

\[ \frac{11}{a + b} + 2(a - b) = 11 \]
and
\[ \frac{22}{a + b} + 3(a - b) = 17, \]

we will follow these steps:

### Step 1: Rewrite the second equation
The second equation can be rewritten as follows:

\[ \frac{22}{a + b} + 3(a - b) = 17. \]

Notice that \(22\) can be expressed as \(2 \times 11\), allowing us to relate it to the first equation.

### Step 2: Substitute variables
Let's substitute:
\[ x = \frac{11}{a + b} \quad \text{and} \quad y = a - b. \]

Now, we can rewrite the equations:
1. From the first equation:
\[ x + 2y = 11 \quad \text{(Equation 1)} \]
2. From the second equation:
\[ 2x + 3y = 17 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of linear equations:
1. \(x + 2y = 11\)
2. \(2x + 3y = 17\)

We can solve these equations using the elimination method.

### Step 4: Multiply Equation 1
Multiply Equation 1 by 2:
\[ 2(x + 2y) = 2(11) \implies 2x + 4y = 22 \quad \text{(Equation 3)} \]

### Step 5: Subtract Equation 2 from Equation 3
Now subtract Equation 2 from Equation 3:
\[ (2x + 4y) - (2x + 3y) = 22 - 17 \]
This simplifies to:
\[ y = 5. \]

### Step 6: Substitute back to find x
Now substitute \(y = 5\) back into Equation 1:
\[ x + 2(5) = 11 \implies x + 10 = 11 \implies x = 1. \]

### Step 7: Substitute back to find a and b
Now we have:
\[ x = \frac{11}{a + b} = 1 \implies a + b = 11, \]
and
\[ y = a - b = 5. \]

### Step 8: Solve for a and b
Now we can solve these two equations:
1. \(a + b = 11\)
2. \(a - b = 5\)

Add these two equations:
\[ (a + b) + (a - b) = 11 + 5 \implies 2a = 16 \implies a = 8. \]

Now substitute \(a = 8\) back into \(a + b = 11\):
\[ 8 + b = 11 \implies b = 11 - 8 \implies b = 3. \]

### Final Solution
Thus, the values of \(a\) and \(b\) are:
\[ a = 8, \quad b = 3. \]