Solve System of Equations: Circle x^2 + y^2 = 25 and Line 2x + y = 10

Math Problem Statement

Solve the following system of equations: 1. x^2 + y^2 = 25 2. 2x + y = 10

The system of equations provided in the image is:

x^2 + y^2 = 25 \\ 2x + y = 10 \end{cases}$$ ### (a) Solve for $$x$$ and $$y$$: 1. From the second equation, solve for $$y$$ in terms of $$x$$: $$y = 10 - 2x$$ 2. Substitute $$y = 10 - 2x$$ into the first equation: $$x^2 + (10 - 2x)^2 = 25$$ 3. Expand $$(10 - 2x)^2$$: $$x^2 + (100 - 40x + 4x^2) = 25$$ 4. Combine like terms: $$5x^2 - 40x + 100 = 25$$ 5. Simplify: $$5x^2 - 40x + 75 = 0$$ 6. Divide through by 5: $$x^2 - 8x + 15 = 0$$ 7. Factorize: $$(x - 5)(x - 3) = 0$$ 8. Solve for $$x$$: $$x = 5 \quad \text{or} \quad x = 3$$ 9. Substitute $$x = 5$$ and $$x = 3$$ into $$y = 10 - 2x$$: - If $$x = 5$$, $$y = 10 - 2(5) = 0$$. - If $$x = 3$$, $$y = 10 - 2(3) = 4$$. Thus, the solutions are: $$(x, y) = (5, 0) \quad \text{and} \quad (x, y) = (3, 4).$$ --- ### (b) Verify graphically: To verify, plot the circle $$x^2 + y^2 = 25$$ and the line $$2x + y = 10$$. The intersection points should match the solutions $$(5, 0)$$ and $$(3, 4)$$. --- Would you like me to: 1. Solve graphically and display the plot? 2. Provide a deeper explanation of any step? 3. Discuss the significance of the graphical verification? 4. Explore what happens if the equations are slightly altered? 5. Solve another system of equations? **Tip:** When solving systems of equations, substitution is often useful when one equation can be easily rearranged.

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Algebra
Systems of Equations
Geometry
Quadratic Equations

Equation of a circle: x^2 + y^2 = r^2
Linear equation: y = mx + b
Factoring: ax^2 + bx + c = 0

Quadratic formula
Graphical intersection of a line and a circle

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