Math Problem Statement
Solve the system of equations: y = x + 1 and xy = 2.
Solution
Let's solve the system of equations using the substitution method. The system is:
y = x + 1, \\ xy = 2. \end{cases}$$ ### Step 1: Substitute $$y = x + 1$$ into $$xy = 2$$ Plugging $$y = x + 1$$ into $$xy = 2$$ gives: $$x(x + 1) = 2.$$ ### Step 2: Form a Quadratic Equation Expanding the left-hand side: $$x^2 + x = 2.$$ Rearrange to form a standard quadratic equation: $$x^2 + x - 2 = 0.$$ ### Step 3: Solve the Quadratic Equation To solve $$x^2 + x - 2 = 0$$, use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$ where $$a = 1$$, $$b = 1$$, and $$c = -2$$. Plugging in the values: $$x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1}.$$ Simplify the discriminant: $$x = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm \sqrt{9}}{2}.$$ $$x = \frac{-1 \pm 3}{2}.$$ Thus, the two solutions for $$x$$ are: $$x_1 = \frac{-1 - 3}{2} = -2, \quad x_2 = \frac{-1 + 3}{2} = 1.$$ ### Step 4: Find Corresponding $$y$$-Values Using $$y = x + 1$$: 1. For $$x_1 = -2$$, $$y_1 = -2 + 1 = -1$$. 2. For $$x_2 = 1$$, $$y_2 = 1 + 1 = 2$$. ### Solutions The solutions to the system are: $$x_1 = -2, \quad y_1 = -1, \quad \text{and} \quad x_2 = 1, \quad y_2 = 2.$$ ### Final Answer $$\begin{aligned} x_1 &= -2, \quad y_1 = -1, \\ x_2 &= 1, \quad y_2 = 2. \end{aligned}$$ --- Would you like a detailed explanation of any step? ### Related Questions 1. How do you solve systems of equations using the substitution method? 2. What is the quadratic formula, and how is it derived? 3. How can you check if your solutions to a system are correct? 4. What types of systems of equations are best solved by substitution? 5. How do different methods of solving systems (substitution, elimination, graphing) compare? **Tip:** Always check your solutions by substituting them back into the original equations to verify correctness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Substitution Method
Quadratic Equations
Formulas
Substitution method
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Substitution method
Quadratic formula
Suitable Grade Level
Grades 8-10
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