Find the values of (x+y ) , if
(i) 3x + 4y = 11 , 4x + 3y = 10 `
(ii) ` 5x - 2y = 4 , x + 8y = 26 `

Answer

To solve the given equations and find the values of \(x + y\), we will follow these steps for both parts of the question.

### Part (i): Solve the equations \(3x + 4y = 11\) and \(4x + 3y = 10\)

1. **Multiply the equations to align coefficients of \(x\)**:
- Multiply the first equation by 4:
\[ 4(3x + 4y) = 4(11) \implies 12x + 16y = 44 \]
- Multiply the second equation by 3:
\[ 3(4x + 3y) = 3(10) \implies 12x + 9y = 30 \]

2. **Set up the new equations**:
- We now have:
\[ 12x + 16y = 44 \quad (1) \]
\[ 12x + 9y = 30 \quad (2) \]

3. **Subtract the second equation from the first**:
\[ (12x + 16y) - (12x + 9y) = 44 - 30 \]
This simplifies to:
\[ 7y = 14 \]

4. **Solve for \(y\)**:
\[ y = \frac{14}{7} = 2 \]

5. **Substitute \(y\) back into one of the original equations to find \(x\)**:
Using the first equation:
\[ 3x + 4(2) = 11 \]
Simplifying gives:
\[ 3x + 8 = 11 \implies 3x = 3 \implies x = 1 \]

6. **Find \(x + y\)**:
\[ x + y = 1 + 2 = 3 \]

### Part (ii): Solve the equations \(5x - 2y = 4\) and \(x + 8y = 26\)

1. **Multiply the second equation to align coefficients of \(x\)**:
- Multiply the second equation by 5:
\[ 5(x + 8y) = 5(26) \implies 5x + 40y = 130 \]

2. **Set up the new equations**:
- We now have:
\[ 5x - 2y = 4 \quad (1) \]
\[ 5x + 40y = 130 \quad (2) \]

3. **Subtract the first equation from the second**:
\[ (5x + 40y) - (5x - 2y) = 130 - 4 \]
This simplifies to:
\[ 42y = 126 \]

4. **Solve for \(y\)**:
\[ y = \frac{126}{42} = 3 \]

5. **Substitute \(y\) back into one of the original equations to find \(x\)**:
Using the first equation:
\[ 5x - 2(3) = 4 \]
Simplifying gives:
\[ 5x - 6 = 4 \implies 5x = 10 \implies x = 2 \]

6. **Find \(x + y\)**:
\[ x + y = 2 + 3 = 5 \]

### Final Answers:
- For part (i), \(x + y = 3\)
- For part (ii), \(x + y = 5\)