EQUALITY OF COMPLEX NUMBERS WORKSHEET

(1)  Find real numbers x and y such that

(x + yi) (2 - i) = ¡

(2)  Find real numbers x and y such that

(3 + 2i) (x + yi)  =  -i

(3)  Find real numbers x and y such that

(x + 2i) (y - i)  =  -4 - 7i

Equality of complex numbers worksheet - Solution

Question 1 :

Find real numbers x and y such that

(x + yi) (2 - i) = ¡

Solution :

(x + yi) (2 - i) = ¡

By multiplying two complex numbers on the left side, we get

2x - ix + 2iy - i²y = ¡

Here the value of i² is -1

(2x + y) + i(2y - x)  =  i

By equating the real and imaginary parts

Applying the value of y in the second equation

Question 2 :

Find real numbers x and y such that

(3 + 2i) (x + yi)  =  -i

Solution :

(3 + 2i) (x + yi)  =  -i

3x + 3iy + 2ix - 2yi²  =  -i

3x + 3iy + 2ix - 2y(-1)  =  -i

(3x + 2y) + i(2x + 3y)  =  -i

Equating the real and imaginary parts.

Applying the value of y in the second equation

2x + 3(-3x/2)  =  -1

2x - (9x/2)  =  -1

(4x - 9x)/2  =  -1

Multiply both sides by 2

-5x  =  -2

Divide -5 on both sides

x  =  2/5

y =  -3(2/5)/2

  =  -(6/5) ⋅ (1/2)

  y  =  -3/5

Hence the value of x  =  2/5 and y  =  -3/5.

Question 3 :

Find real numbers x and y such that

(x + 2i) (y - i)  =  -4 - 7i

Solution :

(x + 2i) (y - i)  =  -4 - 7i

xy - ix + i2y - 2i²  =  -4 - 7i

xy - ix + i2y - 2(-1)  =  -4 - 7i

(xy + 2)- i(x - 2y)  =  -4 - 7i

Applying the value of x in the first equation, we get

(7 + 2y) y +  2  =  -4

7y + 2y² + 2  =  -4

 2y² + 7y + 2 + 4  =  0

 2y² + 7y + 6  =  0

 2y² + 4y + 3y + 6  =  0

 2y(y + 2) + 3(y + 2)  =  0

(2y + 3) (y + 2)  =  0

Hence the solutions are 

x  =  4 and x  =  3

y  =  -3/2 and y  = -2

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