6. If x and y are real numbers such that x+ y√5=1/1-√5 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑥 𝑎𝑛𝑑 𝑦.

We are given the equation:

\[
x + y\sqrt{5} = \frac{1}{1 - \sqrt{5}}
\]

We need to solve for \( x \) and \( y \). The first step is to simplify the right-hand side by rationalizing the denominator.

### Step 1: Rationalize the denominator

We rationalize the denominator by multiplying both the numerator and denominator by the conjugate of \( 1 - \sqrt{5} \), which is \( 1 + \sqrt{5} \):

\[
\frac{1}{1 - \sqrt{5}} \cdot \frac{1 + \sqrt{5}}{1 + \sqrt{5}} = \frac{1 + \sqrt{5}}{(1 - \sqrt{5})(1 + \sqrt{5})}
\]

Now, simplify the denominator:

\[
(1 - \sqrt{5})(1 + \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4
\]

Thus, the expression becomes:

\[
\frac{1 + \sqrt{5}}{-4} = -\frac{1}{4} - \frac{\sqrt{5}}{4}
\]

### Step 2: Compare real and irrational parts

Now we have:

\[
x + y\sqrt{5} = -\frac{1}{4} - \frac{\sqrt{5}}{4}
\]

Compare the real and irrational parts on both sides. On the left-hand side, the real part is \( x \) and the coefficient of \( \sqrt{5} \) is \( y \). On the right-hand side, the real part is \( -\frac{1}{4} \) and the coefficient of \( \sqrt{5} \) is \( -\frac{1}{4} \). Therefore, we can set up the following system of equations:

\[
x = -\frac{1}{4}
\]
\[
y = -\frac{1}{4}
\]

### Final Answer:

Thus, the values of \( x \) and \( y \) are:

\[
x = -\frac{1}{4}, \quad y = -\frac{1}{4}
\]

Would you like more details, or have any questions about the process?

Here are 5 related questions to explore:

1. How do you rationalize a denominator involving square roots?
2. Can you solve \( x + y\sqrt{2} = \frac{2}{1 - \sqrt{2}} \)?
3. What are conjugates of binomial expressions, and why are they useful?
4. How do you compare real and irrational parts in a complex or irrational equation?
5. What happens when you multiply two conjugates involving cube roots?

**Tip**: When rationalizing denominators, always look for the conjugate to simplify the expression.

If x and y are real numbers such that x + y√5 = 1 / (1 - √5), then find x and y.

Find the values of x and y where x + y√5 equals 1 / (1 - √5). This solution involves rationalizing the denominator and comparing real and irrational parts. Ideal for students in Grades 9-11.

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