Video Transcript
Solve the equation 𝑧
multiplied by two plus 𝑖 equals three minus 𝑖 for 𝑧.
To solve this equation for 𝑧,
we’ll need to apply inverse operations. We’ll begin by dividing both
sides of this equation by two plus 𝑖. And we see that 𝑧 is equal to
three minus 𝑖 divided by two plus 𝑖. To divide three minus 𝑖 by two
plus 𝑖, we’re going to need to multiply both the numerator and the denominator
of the fraction by the conjugate of two plus 𝑖. To find the conjugate, we
change the sign of the imaginary part. And we see that the conjugate
of two plus 𝑖 is two minus 𝑖.
We’ll distribute the
parentheses at the top of this fraction by using the FOIL method. Three multiplied by two is
six. Three multiplied by negative 𝑖
is negative three 𝑖. We then get negative two
𝑖. And our last term gives us 𝑖
squared. 𝑖 squared is of course
negative one. So our last term is negative
one.
And we can collect like terms
or add the real parts and separately add the imaginary parts. And we see that three minus 𝑖
multiplied by two minus 𝑖 is five minus five 𝑖. And we could repeat this
process for the denominator.
However, there is a special
rule we can use to multiply a complex number by its conjugate. We can find the sum of the
squares of the real and imaginary parts. The real part is two, and the
imaginary part, the coefficient of 𝑖, is one. So the product of these two
complex numbers is four plus one, which is five. And we see that 𝑧 is equal to
five minus five 𝑖 over five.
We can then divide the real
parts by this real number. We get five divided by five,
which is one. And separately we divide the
imaginary part by this real number. Five divided by five is
one. So we get one minus 𝑖. And we’ve solved our equation
for 𝑧.
