Math Problem Statement
Solve the system of equations: x - y = 2 and x^2 - 2y = 12.
Solution
The system of equations given in the image is:
x - y = 2 \\ x^2 - 2y = 12 \end{cases}$$ To solve this system, let’s follow these steps: ### Step 1: Solve for $$y$$ in the first equation From $$x - y = 2$$, we can express $$y$$ as: $$y = x - 2$$ ### Step 2: Substitute $$y = x - 2$$ into the second equation Substitute $$y$$ into $$x^2 - 2y = 12$$: $$x^2 - 2(x - 2) = 12$$ Simplify: $$x^2 - 2x + 4 = 12$$ $$x^2 - 2x - 8 = 0$$ ### Step 3: Solve the quadratic equation The quadratic equation is: $$x^2 - 2x - 8 = 0$$ Factorize (if possible) or use the quadratic formula: $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)}$$ $$x = \frac{2 \pm \sqrt{4 + 32}}{2}$$ $$x = \frac{2 \pm \sqrt{36}}{2}$$ $$x = \frac{2 \pm 6}{2}$$ $$x = 4 \quad \text{or} \quad x = -2$$ ### Step 4: Find $$y$$ for each $$x$$ Using $$y = x - 2$$: - When $$x = 4$$: $$y = 4 - 2 = 2$$ - When $$x = -2$$: $$y = -2 - 2 = -4$$ ### Final Solution: The solutions are: $$(x, y) = (4, 2) \quad \text{and} \quad (x, y) = (-2, -4)$$ Would you like me to verify the solutions or explain any specific step in more detail? --- #### Related Questions: 1. How do you derive the quadratic formula for general use? 2. Can this system of equations be solved graphically? 3. How do you check the correctness of the solutions? 4. What are the implications of having two solutions for such a system? 5. Can this system be solved using matrices or elimination methods? #### Tip: Always substitute your solutions back into the original equations to verify their correctness!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Quadratic Equations
Formulas
Expression for solving a linear equation: y = mx + b
Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
Theorems
Quadratic Formula
Substitution Method
Suitable Grade Level
Grades 9-10