Question

solve the following simultaneous equations.

ex.
\\(\

$$\begin{cases} xy + x - y = 1 & \\dots\\dots (1) \\\\ xy - 3x + y = 5 & \\dots\\dots (2) \\end{cases}$$

\\)

sol from (1) - (2),
\\4x - 2y = -4\\
therefore, \\(y = 2x + 2 \dots\dots (3)\\)
substituting (3) into (1),
\\x(2x + 2) + x - (2x + 2) = 1\\
\\2x^2 + x - 3 = 0\\
\\(2x + 3)(x - 1) = 0\\
therefore,
\\x = -\frac{3}{2}, 1\\

substituting \\(x = -\frac{3}{2}\\) into (3),
\\y = -1\\
substituting \\(x = 1\\) into (3),
\\y = 4\\

ans. \\(\

$$\begin{cases} x = -\\frac{3}{2} \\\\ y = -1 \\end{cases}$$

\\), \\(\

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

\\)

(1) \\(\

$$\begin{cases} xy + 3x - 3y + 3 = 0 \\\\ xy - x - y + 1 = 0 \\end{cases}$$

\\)

Explanation:

Step1: Write down the equations

$$
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$$

Step2: Subtract equation (2) from (1)

$$4x - 2y + 2 = 0$$

Step3: Express y in terms of x

$$y = 2x + 1 \quad \cdots \text{(3)}$$

Step4: Substitute (3) into (2)

$$x(2x + 1) - x - (2x + 1) + 1 = 0$$

Step5: Simplify the quadratic equation

$$2x^2 - 2x = 0$$

Step6: Solve for x

$$2x(x - 1) = 0 \implies x = 0 \text{ or } x = 1$$

Step7: Find corresponding y values

$$\text{If } x = 0, y = 1; \text{ if } x = 1, y = 3$$

Answer:

$$
LATEXBLOCK0
\text{ or }
LATEXBLOCK1
$$