Question

j147a kumon
simultaneous equations

solve the following equations.

ex.
\\(\

$$\begin{cases} x + y = 9 & \\dots\\dots \\text{(1)} \\\\ xy = 7 & \\dots\\dots \\text{(2)} \\end{cases}$$

\\)

sol \\(x\\) and \\(y\\) are the roots of
\\t^2 - 9t + 7 = 0\\
therefore, \\(t = \frac{9 \pm \sqrt{53}}{2}\\)

ans. \\(\

$$\begin{cases} x = \\frac{9 + \\sqrt{53}}{2} \\\\ y = \\frac{9 - \\sqrt{53}}{2} \\end{cases}$$

\\), \\(\

$$\begin{cases} x = \\frac{9 - \\sqrt{53}}{2} \\\\ y = \\frac{9 + \\sqrt{53}}{2} \\end{cases}$$

\\)

(1) \\(\

$$\begin{cases} x + y = 4 \\\\ xy = 2 \\end{cases}$$

\\)

alternative method for example: from \text{(1)}, \\(y = 9 - x\\). we can substitute this into \text{(2)}, and solve as usual. however, with terms such as \\(x + y\\) and \\(xy\\), it is easier to use the root-coefficient relationships.

Explanation:

Step1: Formulate the quadratic equation

$$ t^2 - 4t + 2 = 0 $$

Step2: Apply the quadratic formula

$$ t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(2)}}{2(1)} $$

Step3: Simplify the expression

$$ t = 2 \pm \sqrt{2} $$

Step4: State the solutions for x and y

$$
LATEXBLOCK0
\quad \text{or} \quad
LATEXBLOCK1
$$

Answer:

$$
LATEXBLOCK0
\quad \text{or} \quad
LATEXBLOCK1
$$