Question

j147b solve the following simultaneous equations

(2) \\(\

$$\begin{cases} x + y = 5 \\\\ xy = -6 \\end{cases}$$

\\)

(3) \\(\

$$\begin{cases} x + y = \\sqrt{3} + 1 \\\\ xy = \\sqrt{3} \\end{cases}$$

\\)

Explanation:

Step1: Express y in terms of x for (2)

$$y = 5 - x$$

Step2: Substitute y into the second equation of (2)

$$x(5 - x) = -6$$

Step3: Solve the quadratic equation for x

$$x^2 - 5x - 6 = 0 \implies (x - 6)(x + 1) = 0 \implies x = 6 \text{ or } x = -1$$

Step4: Find corresponding y values for (2)

$$\text{If } x = 6, y = -1; \text{ if } x = -1, y = 6$$

Step5: Express y in terms of x for (3)

$$y = \sqrt{3} + 1 - x$$

Step6: Substitute y into the second equation of (3)

$$x(\sqrt{3} + 1 - x) = \sqrt{3}$$

Step7: Solve the quadratic equation for x

$$x^2 - (\sqrt{3} + 1)x + \sqrt{3} = 0 \implies (x - \sqrt{3})(x - 1) = 0 \implies x = \sqrt{3} \text{ or } x = 1$$

Step8: Find corresponding y values for (3)

$$\text{If } x = \sqrt{3}, y = 1; \text{ if } x = 1, y = \sqrt{3}$$

Answer:

(2) \(x = 6, y = -1\) or \(x = -1, y = 6\)
(3) \(x = \sqrt{3}, y = 1\) or \(x = 1, y = \sqrt{3}\)