Question

use the substitution method to find all solutions of the system
\\\

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

\\
the solutions of the system are:
\\x_1 = \text{___}, y_1 = \text{___}\\ and
\\x_2 = \text{___}, y_2 = \text{___}\\ with \\(x_1 < x_2\\).

Explanation:

Step1: Substitute the first equation into the second

Substitute \(y = x - 1\) into \(xy = 2\):

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

Step2: Solve the resulting quadratic equation

Expand and rearrange the equation into standard form:

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

Factor the quadratic equation:

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

Thus, the solutions for \(x\) are:

$$x = 2 \quad \text{or} \quad x = -1$$

Step3: Find the corresponding y-values

Since \(x_1 < x_2\), we set \(x_1 = -1\) and \(x_2 = 2\).
For \(x_1 = -1\):

$$y_1 = -1 - 1 = -2$$

For \(x_2 = 2\):

$$y_2 = 2 - 1 = 1$$

Answer:

\(x_1 = -1\), \(y_1 = -2\) and \(x_2 = 2\), \(y_2 = 1\)