X Y X \ Y

Algebra Level 2

( 1 + i ) ( x − y i ) = i ( 14 + 7 i ) − ( 2 + 13 i ) \large (1+i)(x-yi) = i(14+7i)-(2+13i)

Find x x and y y .

Notation: i = − 1 i = \sqrt{-1} denotes the imaginary unit .

x = − 4 , y = − 5 x=-4, \ y=-5 x = 5 , y = 4 x=5, \ y=4 x = − 2 , y = − 4 x=-2, \ y=-4 x = 5 , y = − 3 x=5, \ y=-3

Clent Pacilan by Clent Pacilan , 17, Philippines

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1 solution

( 1 + i ) ( x − y i ) = i ( 14 + 7 i ) − ( 2 + 13 i ) x − y i + x i + y = 14 i − 7 − 2 − 13 i ( x + y ) + ( x − y ) i = − 9 + i \begin{aligned} (1+i)(x-yi) & = i(14+7i) - (2+13i) \\ x-yi +xi + y & = 14i -7 - 2- 13i \\ (x+y) +(x-y)i & = -9 + i \end{aligned}

Equating the real and imaginary parts on both sides,

{ x + y = − 9 . . . ( 1 ) x − y = 1 . . . ( 2 ) \begin{cases} x + y = -9 & ...(1) \\ x-y = 1 & ...(2) \end{cases} ⟹ ( 1 ) + ( 2 ) : 2 x = − 8 ⟹ x = − 4 ⟹ ( 1 ) : − 4 + y = − 9 ⟹ y = − 5 \implies (1)+(2): 2x = - 8 \implies x = -4 \implies (1): -4+y =-9 \implies y = -5 .

Therefore, the answer is x = − 4 , y = − 5 \boxed{x=-4, \ y=-5} .

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