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How Do You Find Values for x and y to Make Two Complex Numbers Equal?

Find x and y in 6x + 2yi = 2x + 4(2 + 3i).

Summary

  1. If two complex numbers are equal, then the real parts are equal and the imaginary parts are equal
  2. 'x' and 'y' are variables, and 'i' is the imaginary unit and equals the square root of -1
  3. The first step is to simplify by distributing the 4 into the parentheses
  4. Any term that does not contain 'i' is real and is labeled with 'R'
  5. Any term that contains 'i' is imaginary, and is labeled with 'I'

Notes

    1. If two complex numbers are equal, then the real parts are equal and the imaginary parts are equal
    2. The real part of a complex number is the part WITHOUT an 'i'
    3. The imaginary part of a complex number is the part WITH an 'i'
    4. Remember, 'i' is the imaginary unit and is equal to the square root of -1
    1. 'a', 'b', 'c', and 'd' are coefficients, and 'i' is the imaginary unit
    2. The real part of a complex number is the part WITHOUT an 'i'
    3. The imaginary part of a complex number is the part WITH an 'i'
    4. So the real parts must be equal to each other, and the imaginary parts must be equal to each other
    1. To find 'x' and 'y', we need to set the real parts equal to each other and the imaginary parts equal to each other
    2. But we still have some parentheses on the right hand side that we need to get rid of first
    1. Since we can't combine the real and imaginary parts of a complex number, we can't do anything else with the left hand side
    1. Multiplying 4 through (2 + 3i) gives us 8 + 12i
    2. So we now have a complex number in the form c + di, where c = 2x+8 and d = 12
    3. 'x' is a variable, and 'i' is the imaginary unit equal to the square root of -1
    1. The real part of a complex number is the part WITHOUT an 'i'
    2. The imaginary part of a complex number is the part WITH an 'i'
    3. Remember, 'i' is the imaginary unit and is equal to the square root of -1
    1. Recall that complex numbers have one part that is real and one part that is imaginary
    2. The real part of a complex number is the part WITHOUT an 'i'
    3. The imaginary part of a complex number is the part WITH an 'i'
    4. 'x' and 'y' are variables, and 'i' is the imaginary unit equal to the square root of -1
    5. We can't combine 2x and 8 since they're not like terms, but since neither of them has an 'i' they are both still part of the real part
    1. Recall that the Equality of Complex Numbers says the real parts should be equal and the imaginary parts should be equal
    2. So we can set the real parts equal to each other to solve for 'x'
    3. Our real parts are 6x and 2x+8
    1. Recall that the Equality of Complex Numbers says the real parts should be equal and the imaginary parts should be equal
    2. Our real parts are 6x and 2x+8
    3. Subtracting 2x from 6x on the left leaves us with 4x
    4. Subtracting 2x on the right cancels out the other 2x, leaving us with 8
    1. On the left, the 4s cancel and we get 'x' by itself
    2. On the right, 8 divided by 4 is 2
    1. Recall that the Equality of Complex Numbers says the real parts should be equal and the imaginary parts should be equal
    2. So we can set the imaginary parts equal to each other to solve for 'y'
    3. Our imaginary parts are 2yi and 12i, since they both contain 'i'
    1. Recall that the Equality of Complex Numbers says the real parts should be equal and the imaginary parts should be equal
    2. Our imaginary parts are 2yi and 12i, since they both contain 'i'
    3. Dividing by 2i on the left cancels out the 2 and the 'i', leaving just 'y'
    4. On the right, the 'i's cancel out, and 12 divided by 2 reduces to 6