Solve the system of equations. $x^{2} + y^{2} = 25$, $x - y = 5$
- $(-5, 0)$ only
- $(0, 5)$ only
- $(0, -5)$ and $(-5, 0)$
- $(0, 5)$ and $(5, 0)$
- $(0, -5)$ and $(5, 0)$
Reveal answer
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Correct answer
Explanation
From x - y = 5, x = y + 5. Substitute into x^2 + y^2 = 25: (y + 5)^2 + y^2 = 25. y^2 + 10y + 25 + y^2 = 25. 2y^2 + 10y = 0. 2y(y + 5) = 0. So y = 0 or y = -5. If y = 0, x = 5. If y = -5, x = 0. Solutions are (5, 0) and (0, -5).
AI explanation
From the second equation, express x as x = y + 5. Substituting this into the first equation gives (y + 5)^2 + y^2 = 25. Expanding and simplifying this results in 2y^2 + 10y = 0, which factors to 2y(y + 5) = 0, yielding y = 0 or y = -5. Substituting these values back into x = y + 5 gives the corresponding x values of 5 and 0, making the solutions (0, -5) and (5, 0).