Simultaneous Surds Efc8F6

1. **Problem Statement:** Solve the simultaneous equations involving surds. 2. **General Approach:** To solve simultaneous equations, we use substitution or elimination methods. When surds are involved, treat them as algebraic terms and simplify carefully. 3. **Example Problem:** Solve the system: $$\begin{cases} x + y = 3 \\ x - y = \sqrt{5} \end{cases}$$ 4. **Step 1: Add the two equations to eliminate $y$:** $$ (x + y) + (x - y) = 3 + \sqrt{5} $$ $$ 2x = 3 + \sqrt{5} $$ $$ x = \frac{3 + \sqrt{5}}{2} $$ 5. **Step 2: Substitute $x$ back into the first equation to find $y$:** $$ \frac{3 + \sqrt{5}}{2} + y = 3 $$ $$ y = 3 - \frac{3 + \sqrt{5}}{2} $$ $$ y = \frac{6 - (3 + \sqrt{5})}{2} = \frac{3 - \sqrt{5}}{2} $$ 6. **Final Answer:** $$ x = \frac{3 + \sqrt{5}}{2}, \quad y = \frac{3 - \sqrt{5}}{2} $$ This method can be applied to other simultaneous equations involving surds by carefully isolating variables and simplifying expressions.