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МатематикаВопрос:
Solve the system of equations: \begin{cases} \frac{1}{x} - \frac{1}{y} = \frac{1}{6}, \\ x - y = -1; \end{cases}
Ответ:
Let’s solve the system of equations:
\[\begin{cases} \frac{1}{x} - \frac{1}{y} = \frac{1}{6}, \\ x - y = -1; \end{cases}\]Краткое пояснение: First, we express one variable from the second equation, then substitute it into the first equation and solve for the other variable.
Пошаговое решение:
- Express x from the second equation: \[x - y = -1 \Rightarrow x = y - 1\]
- Substitute x into the first equation: \[\frac{1}{y - 1} - \frac{1}{y} = \frac{1}{6}\]
- Solve for y: \[\frac{y - (y - 1)}{y(y - 1)} = \frac{1}{6}\] \[\frac{1}{y^2 - y} = \frac{1}{6}\] \[y^2 - y = 6\] \[y^2 - y - 6 = 0\]
- Solve the quadratic equation for y: \[y^2 - y - 6 = 0\] \[(y - 3)(y + 2) = 0\] \[y = 3 \text{ or } y = -2\]
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Find the corresponding x values:
- If \(y = 3\): \[x = y - 1 = 3 - 1 = 2\]
- If \(y = -2\): \[x = y - 1 = -2 - 1 = -3\]
Answer: The solutions are \((2, 3)\) and \((-3, -2)\).
