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Video: Solving a Two-Variable Linear Equation with Complex Coefficients

Tim Burnham

Find the real values of 𝑥 and 𝑦 that satisfy the equation (2𝑥 − 5) + 𝑦𝑖 = −3 − 5𝑖.

01:26

Video Transcript

Find the real values of 𝑥 and 𝑦 that satisfy the equation two 𝑥 minus five plus 𝑦𝑖 is equal to negative three take away five 𝑖.

Well our question has given us two complex numbers, one on either side of the equation. And in complex numbers, they’re made up of a real part which is this bit here, and an imaginary part which in this case this bit here. Now the real components of these must be equal, because, well, overall numbers are equal. And the imaginary parts must be equal. So we can actually create two equations.

If the real parts are equal, then two 𝑥 minus five must be equal to negative three. Then if I add five to both sides of that equation, two 𝑥 minus five on the left-hand side is just two 𝑥. And negative three plus five on the right-hand side is positive two. Now if I divide both sides by two, I get 𝑥 is equal to one.

Now also the imaginary parts are equal, so 𝑦 must be equal to negative five. This part of this number is equal to this part of this number. So the answer then 𝑥 equals one and 𝑦 equals negative five.