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System Solution Sum (Posted on 2024-11-18)

x and y are real numbers such that x² + 3y = 9, and 2y² - 6x = 18. If the 4 solutions to the system are (x₁,y₁), (x₂,y₂), (x₃,y₃), and (x₄,y₄), then find (y₁ - x₁) + (y₂ - x₂) + (y₃ - x₃) + (y₄ - x₄).

No Solution Yet Submitted by Danish Ahmed Khan

Rating: 5.0000 (1 votes)

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Solution without solving

Comment 3 of 3 |

The connection between these equations is not very well hidden:

Rewrite them as

x² - 9 = -3y, and y² - 9 = 3x

and note they are transformations of each other. Specifically the transformation is (x,y) -> (-y,-x) which is the reflection over the line y=-x.

Thus any solution (intersection point) is either

1.) on the line y=-x and its coordinates will sum to zero

2.) for any intersection point (a,b) there will also be the reflection point (-b,-a). (a+b)+(-b+-a)=0

So knowing nothing else of the intersections, we can conclude the expression equals 0.

Posted by Jer on 2024-11-18 11:37:47

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