Math Problem Statement

x^(2)+y^(2)=40 (x-5)^(2)+y^(2)=45

Solution

To solve the given system of equations:

x^2 + y^2 = 40 \\ (x - 5)^2 + y^2 = 45 \end{cases}$$ we need to find the values of $$x$$ and $$y$$ that satisfy both equations simultaneously. First, expand the second equation: $$(x - 5)^2 + y^2 = 45$$ Expanding $$(x - 5)^2$$: $$(x - 5)^2 = x^2 - 10x + 25$$ So the second equation becomes: $$x^2 - 10x + 25 + y^2 = 45$$ Now we have two equations: $$\begin{cases} x^2 + y^2 = 40 \\ x^2 - 10x + 25 + y^2 = 45 \end{cases}$$ Subtract the first equation from the second equation: $$(x^2 - 10x + 25 + y^2) - (x^2 + y^2) = 45 - 40$$ This simplifies to: $$-10x + 25 = 5$$ Solving for $$x$$: $$-10x + 25 = 5 \implies -10x = -20 \implies x = 2$$ Now, substitute $$x = 2$$ back into the first equation to solve for $$y$$: $$2^2 + y^2 = 40 \implies 4 + y^2 = 40 \implies y^2 = 36 \implies y = \pm 6$$ So, the solutions are: $$(x, y) = (2, 6) \text{ or } (2, -6)$$ Therefore, the solutions to the system are $$(2, 6)$$ and $$(2, -6)$$. Would you like details on any specific part or have any questions? Here are some related questions for further practice: 1. Solve the system of equations $$x^2 + y^2 = 25$$ and $$(x - 3)^2 + y^2 = 20$$. 2. Find the intersection points of the circles $$x^2 + y^2 = 16$$ and $$(x + 4)^2 + y^2 = 25$$. 3. Determine the coordinates of the points where the ellipse $$x^2 + 4y^2 = 20$$ intersects the line $$x - y = 1$$. 4. Solve the system $$x^2 + y^2 = 49$$ and $$(x - 7)^2 + y^2 = 100$$. 5. Determine the points of intersection for $$x^2 + y^2 = 9$$ and $$x^2 + (y - 4)^2 = 25$$. 6. Find the points where the parabola $$y = x^2 - 4$$ intersects the circle $$x^2 + y^2 = 16$$. 7. Solve the system $$x^2 + y^2 = 13$$ and $$(x - 2)^2 + (y - 3)^2 = 10$$. 8. Determine the intersection points of the circles $$x^2 + y^2 = 10$$ and $$(x + 5)^2 + y^2 = 26$$. **Tip:** When solving systems of nonlinear equations, it often helps to expand and simplify one of the equations before substituting it into the other.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Circle Equations

Formulas

Quadratic equation formula ax^2 + bx + c = 0

Theorems

-

Suitable Grade Level

Grades 10-12

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