Question

solve the following equations.

(1) \\(\

$$\begin{cases} x^2 + 9xy + y^2 = 23 & \\dots\\dots (1) \\\\ xy + x + y = 5 & \\dots\\dots (2) \\end{cases}$$

\\)

sol assuming that \\(x + y = a\\), \\(xy = b\\),
from (1) and (2),
\\(\

$$\begin{cases} a^2 + 7b = 23 & \\dots\\dots (3) \\\\ a + b = 5 & \\dots\\dots (4) \\end{cases}$$

\\)

from (4), \\(b = 5 - a\\)
substituting this into (3),
\\(a^2 + 7( \quad ) = 23\\)
\\(a^2 - \quada + \quad = 0\\)
\\(( \quad )( \quad ) = 0\\)
when \\(a = \quad\\), \\(b = \quad\\)
when \\(a = \quad\\), \\(b = \quad\\)

(i) when \\(\

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

\\)
(ii) when \\(\

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

\\)

Explanation:

Step1: Substitute B into equation (3)

$$A^2 + 7(5 - A) = 23$$

Step2: Expand and simplify to standard form

$$A^2 - 7A + 35 = 23 \implies A^2 - 7A + 12 = 0$$

Step3: Factor the quadratic equation

$$(A - 3)(A - 4) = 0$$

Step4: Solve for A and find corresponding B

$$A = 3 \implies B = 5 - 3 = 2$$

$$A = 4 \implies B = 5 - 4 = 1$$

Step5: Write the systems for x and y

$$\text{(i) }
LATEXBLOCK0
\quad \text{(ii) }
LATEXBLOCK1
$$

Answer:

The completed steps for the solution are:

Substituting \(B = 5 - A\) into (3):
\(A^2 + 7(5 - A) = 23\)

Simplifying the equation:
\(A^2 - 7A + 12 = 0\)

Factoring the quadratic equation:
\((A - 3)(A - 4) = 0\)

Finding the values of \(A\) and \(B\):
When \(A = 3\), \(B = 2\)
When \(A = 4\), \(B = 1\)

Substituting back to \(x\) and \(y\):
(i) When \(

$$\begin{cases} x + y = 3 \\ xy = 2 \end{cases}$$

\)
(ii) When \(

$$\begin{cases} x + y = 4 \\ xy = 1 \end{cases}$$

\)