Solve the system of equations: { x - y = -2 { x - 2y = 4

Решение системы уравнений:

У нас есть система из двух уравнений:

• \[ x - y = -2 \quad (1) \]
• \[ x - 2y = 4 \quad (2) \]

Шаг 1: Выразим x из первого уравнения.

• Из уравнения (1) получаем:
• \[ x = y - 2 \]

Шаг 2: Подставим полученное выражение для x во второе уравнение.

• Подставляем y - 2 вместо x в уравнение (2):
• \[ (y - 2) - 2y = 4 \]

Шаг 3: Решим полученное уравнение относительно y.

• \[ y - 2 - 2y = 4 \]
• \[ -y - 2 = 4 \]
• \[ -y = 4 + 2 \]
• \[ -y = 6 \]
• \[ y = -6 \]

Шаг 4: Найдем значение x, подставив найденное значение y в выражение для x.

• \[ x = y - 2 \]
• \[ x = -6 - 2 \]
• \[ x = -8 \]

Шаг 5: Проверка.

• Подставим найденные значения x = -8 и y = -6 в исходные уравнения.
• Уравнение (1): The original image contains a system of two linear equations: x - y = -2 and x - 2y = 4. The task is to solve this system. This is a standard algebra problem that can be solved using methods such as substitution or elimination. The context provided by the system instruction indicates that the response should be in JSON format, with 'question' and 'answer' fields, and that mathematical formulas should be rendered using MathJax. The problem falls under 'STEM' subjects, requiring a detailed, step-by-step explanation suitable for a younger audience.
  • Analyze the input: The input is a system of two linear equations.
  • Determine the subject: Mathematics (Algebra).
  • Determine the class level: Middle school or early high school (typically grades 7-9).
  • Choose the appropriate protocol: Protocol 3.1 (Exact Sciences).
  • Break down the problem: The system can be solved by substitution or elimination. Substitution is a good method for this particular system.
  • Step 1: Isolate one variable in one equation. From the first equation, $$x - y = -2$$, we can easily isolate $$x$$ by adding $$y$$ to both sides: $$x = y - 2$$.
  • Step 2: Substitute this expression into the other equation. Substitute $$y - 2$$ for $$x$$ in the second equation, $$x - 2y = 4$$: $$(y - 2) - 2y = 4$$.
  • Step 3: Solve for the remaining variable. Simplify and solve the equation for $$y$$: $$y - 2 - 2y = 4 ightarrow -y - 2 = 4 ightarrow -y = 6 ightarrow y = -6$$.
  • Step 4: Substitute the found value back into the isolated expression to find the other variable. Substitute $$y = -6$$ into $$x = y - 2$$: $$x = -6 - 2 ightarrow x = -8$$.
  • Step 5: Verify the solution. Plug $$x = -8$$ and $$y = -6$$ back into the original equations:
  • Equation 1: $$-8 - (-6) = -8 + 6 = -2$$ (Correct)
  • Equation 2: $$-8 - 2(-6) = -8 + 12 = 4$$ (Correct)
  • Format the output: The output should be a JSON object with 'question' and 'answer' fields. The 'answer' field must contain HTML with MathJax for formulas and a friendly, step-by-step explanation.