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Given equations are

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x² + y² = 100 3x - y = 10

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Apply square both sides.

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(3 x - y)² = 10²

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9x² + y² - 6xy = 100

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8x² + x² + y² - 6xy - 100 = 0

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Substitute : x² + y² = 100

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8x² +100 - 6xy - 100 = 0

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8x² - 6xy = 0

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Substitute : 3 x - y = 10 ⇒ y = 3x - 10

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8x² - 6x(3x - 10) = 0

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8x² - 18x² + 60x = 0

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- 10x² + 60x = 0

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60x - 10x² = 0

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10x( 6 - x) = 0

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By using zero product property : If AB = 0 then A = 0 , B = 0.

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10x  = 0  and  ( 6 - x) = 0

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x = 0  and   x = 6

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If x = 0

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3 x - y = 10 ⇒ y = - 10

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If x = 6

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3 x - y = 10 ⇒ y = 10 - 3*6 = - 8

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Solution :

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Given equations are two solutions

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1) x = 0 , y = - 10

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2) x = 6 , y = - 8

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