Solve `{:(13x + 11y = 70),(11x + 13y = 74):}`

Answer

To solve the system of equations:

1. **Equations Given:**
\[ 13x + 11y = 70 \quad \text{(Equation 1)} \]
\[ 11x + 13y = 74 \quad \text{(Equation 2)} \]

2. **Adding the Equations:**
We will add both equations to eliminate one variable.
\[ (13x + 11y) + (11x + 13y) = 70 + 74 \]
Simplifying the left side:
\[ 24x + 24y = 144 \]
Dividing the entire equation by 24:
\[ x + y = 6 \quad \text{(Equation 3)} \]

3. **Subtracting Equation 2 from Equation 1:**
Now, we will subtract Equation 2 from Equation 1.
\[ (13x + 11y) - (11x + 13y) = 70 - 74 \]
Simplifying the left side:
\[ 2x - 2y = -4 \]
Dividing the entire equation by 2:
\[ x - y = -2 \quad \text{(Equation 4)} \]

4. **Adding Equations 3 and 4:**
Now we will add Equation 3 and Equation 4.
\[ (x + y) + (x - y) = 6 - 2 \]
Simplifying:
\[ 2x = 4 \]
Dividing by 2:
\[ x = 2 \]

5. **Substituting x back to find y:**
Now we will substitute \( x = 2 \) back into Equation 3 to find y.
\[ 2 + y = 6 \]
Subtracting 2 from both sides:
\[ y = 4 \]

6. **Final Solution:**
The solution to the system of equations is:
\[ x = 2, \quad y = 4 \]