Question

j 148a kumon
simultaneous equations

solve the following equations.

ex.
\\(\

$$\begin{cases} x + y = 4 & \\dots\\dots (1) \\\\ x^2 + xy + y^2 = 13 & \\dots\\dots (2) \\end{cases}$$

\\)

sol from (2),
\\((x + y)^2 - xy = 13 \dots\dots (3)\\)
substituting (1) into (3),
\\(16 - xy = 13\\)
\\(xy = 3 \dots\dots (4)\\)
from (1) and (4), \\(x\\) and \\(y\\) are the roots of
\\(t^2 - 4t + 3 = 0\\)
\\((t - 3)(t - 1) = 0\\)
therefore, \\(t = 3, 1\\)

ans. \\(\

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

\\), \\(\

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

\\)

(1) \\(\

$$\begin{cases} x + y = 4 \\\\ x^2 - xy + y^2 = 22 \\end{cases}$$

\\)

Explanation:

Step1: Rewrite the second equation

$$ (x+y)^2 - 3xy = 22 $$

Step2: Substitute the first equation

$$ 4^2 - 3xy = 22 $$

Step3: Solve for the product

$$ xy = -2 $$

Step4: Form the quadratic equation

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

Step5: Solve for the roots

$$ t = 2 \pm \sqrt{6} $$

Step6: State the final solutions

$$
LATEXBLOCK0
, \quad
LATEXBLOCK1
$$

Answer:

$$
LATEXBLOCK0
, \quad
LATEXBLOCK1
$$