Math Problem Statement
You are given the following system of equations. Solve the system by substitution or by a graphical method, and explain how to choose a method:
y = x + 1
y^2 + x^2 = 25
Solution
The system of equations presented is:
y = x + 1 \\ y^2 + x^2 = 25 \end{cases}$$ ### Step-by-Step Solution Using Substitution: 1. **Substitute $$y = x + 1$$ into the second equation**: The second equation is: $$y^2 + x^2 = 25$$ Substitute $$y = x + 1$$: $$(x + 1)^2 + x^2 = 25$$ 2. **Simplify the equation**: Expand $$(x + 1)^2$$: $$x^2 + 2x + 1 + x^2 = 25$$ Combine like terms: $$2x^2 + 2x + 1 = 25$$ Subtract 25 from both sides: $$2x^2 + 2x - 24 = 0$$ Simplify by dividing by 2: $$x^2 + x - 12 = 0$$ 3. **Solve the quadratic equation** $$x^2 + x - 12 = 0$$: Factor the quadratic: $$(x + 4)(x - 3) = 0$$ Thus, the solutions for $$x$$ are: $$x = -4 \quad \text{and} \quad x = 3$$ 4. **Find the corresponding $$y$$-values**: Using $$y = x + 1$$: - For $$x = -4$$: $$y = -4 + 1 = -3$$ - For $$x = 3$$: $$y = 3 + 1 = 4$$ 5. **Solutions**: The system has two solutions: $$(-4, -3) \quad \text{and} \quad (3, 4)$$ ### Method Choice Explanation - **Substitution** was chosen because the first equation is already solved for $$y$$, making it easy to substitute into the second equation. - A **graphical method** would also be useful to visually confirm the intersection points of the circle $$y^2 + x^2 = 25$$ and the line $$y = x + 1$$. Would you like more details or have any other questions? Here are 5 related questions for further practice: 1. What is the solution of the system if the line equation was $$y = 2x + 1$$ instead? 2. How do you graphically find the intersection points of a circle and a line? 3. Can you describe the nature of the solutions when a line intersects a circle at exactly one point? 4. What happens if the system involves an ellipse instead of a circle? 5. How does changing the radius of the circle affect the solutions? **Tip**: Always check your solutions by substituting 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
Substitution
Quadratic Equations
Analytic Geometry
Formulas
Substitution Method
Quadratic equation formula: ax^2 + bx + c = 0
Equation of a circle: x^2 + y^2 = r^2
Theorems
Quadratic Formula
Properties of Circle
Methods of Solving Systems of Equations
Suitable Grade Level
Grades 9-11
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