### Video Transcript

Solve two π§ minus π§ bar equals
five in the set of complex numbers.

In this question, we have an
equation involving a complex number π§. We would say in general that π§ is
of the form π plus ππ, where π and π are real constants. π§ bar, which is sometimes written
as π§ star, is the complex conjugate of π§. And we find this by changing the
sign of the imaginary parts. So, the conjugate of our general
complex number π plus ππ is π minus ππ.

Letβs substitute these complex
numbers into our original equation. When we do, we get two times π
plus ππ minus π minus ππ equals five. Weβre now going to distribute our
parentheses. Two times π is two π, and two
times ππ is two ππ. Negative one times π is negative
π, and negative one times negative ππ is positive ππ. So, our equation is two π plus two
ππ minus π plus ππ equals five.

Letβs collect like terms. We get two π minus π, which is
π, and two ππ plus ππ, which is three ππ. So, we get π plus three ππ
equals five. What weβre going to do next is
equate the real and imaginary parts of both sides of our equation. On the left-hand side, the real
part is π. And on the right, the real part is
five. On the left, the imaginary part is
three π. Remember, thatβs the coefficient of
π. And we could say that on the right,
the imaginary part is zero.

By equating the real parts, we find
π is equal to five. Then, by equating the imaginary
parts, we get three π is equal to zero. But this, of course, means that π
itself is equal to zero. So, going back to that general form
of a complex number, π plus ππ, we can say that π§ must be equal to five plus
zero π, which is simply five.