Math Problem Statement
Solve the following system by the addition method.
Solution
To solve the system of equations using the addition method (also known as the elimination method), we first write the system again:
x^2 + y^2 = 40 \\ (x - 5)^2 + y^2 = 45 \end{cases}$$ We already expanded and simplified the second equation in the previous step: 1. $$x^2 + y^2 = 40$$ 2. $$x^2 - 10x + 25 + y^2 = 45$$ Now, we subtract equation 1 from equation 2 to eliminate $$y^2$$: $$(x^2 - 10x + 25 + y^2) - (x^2 + y^2) = 45 - 40$$ Simplifying this, we get: $$-10x + 25 = 5$$ Solving for $$x$$: $$-10x + 25 = 5 \implies -10x = -20 \implies x = 2$$ Now that we have $$x = 2$$, we substitute this value back into equation 1 to find $$y$$: $$2^2 + y^2 = 40 \implies 4 + y^2 = 40 \implies y^2 = 36 \implies y = \pm 6$$ Thus, the solutions are: $$(x, y) = (2, 6) \text{ or } (2, -6)$$ Therefore, the solutions to the system are $$(2, 6)$$ and $$(2, -6)$$. Would you like further details on any specific part or have any additional questions? Here are some related questions for further practice: 1. Solve the system of equations $$x^2 + y^2 = 25$$ and $$(x - 3)^2 + y^2 = 20$$ using the addition method. 2. Find the intersection points of the circles $$x^2 + y^2 = 16$$ and $$(x + 4)^2 + y^2 = 25$$ by elimination. 3. Use the addition method to solve the system $$x^2 + y^2 = 49$$ and $$(x - 7)^2 + y^2 = 100$$. 4. Solve the system $$x^2 + y^2 = 9$$ and $$x^2 + (y - 4)^2 = 25$$ using the elimination method. 5. Determine the intersection points for the equations $$x^2 + y^2 = 13$$ and $$(x - 2)^2 + (y - 3)^2 = 10$$ by elimination. 6. Find the points where the circles $$x^2 + y^2 = 10$$ and $$(x + 5)^2 + y^2 = 26$$ intersect using the addition method. 7. Solve the system of nonlinear equations $$x^2 + y^2 = 30$$ and $$(x + 3)^2 + y^2 = 49$$ using elimination. 8. Use the addition method to solve $$x^2 + y^2 = 20$$ and $$(x - 4)^2 + y^2 = 36$$. **Tip:** When using the addition (elimination) method, carefully choose which variable to eliminate first to simplify the equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Equations
Quadratic Equations
Elimination Method
Formulas
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Theorems
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Suitable Grade Level
High School
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