### Video Transcript

Determine the real numbers π₯ and
π¦ that satisfy the equation five π₯ plus two plus three π¦ minus five π equals
negative three plus four π.

Letβs look carefully at what weβve
been given. We have been given two complex
numbers that weβre told are equal to each other. Now I know it doesnβt look like it,
but that expression to the left of the equal sign is indeed a complex number. Remember, a complex number is one
of the form π plus ππ, where π and π are real numbers.
And weβre told that π₯ and
~~π~~
[π¦] are
real numbers.
And this means that the expression
five π₯ plus two must be real and three π¦ minus five must be real. So five π₯ plus two plus three π¦
minus five π is a complex number. It has a real part of five π₯ plus
two and an imaginary part of three π¦ minus five.

Next, weβll recall what it actually
means for two complex numbers to be equal. We see that two complex numbers π
plus ππ and π plus ππ are equal if π is equal to π and π is equal to π. In other words, their real parts
must be equal and their imaginary parts must separately be equal.

Letβs begin with the real parts in
our question. We saw that the real part of the
complex number on the left is five π₯ plus two. And on the right, itβs negative
three. This means that five π₯ plus two
must be equal to negative three. Weβll solve this as normal by
applying a series of inverse operations. Weβll subtract two from both sides,
and then weβll divide through by five. And we see that π₯ is equal to
negative one.

Letβs repeat this process for the
imaginary parts. We said that the imaginary part for
our number on the left is three π¦ minus five. And on the right, we can see itβs
four. This means that three π¦ minus five
must be equal to four. We can add five to both sides of
this equation. And then weβll divide through by
three. And we see that π¦ must be equal to
three. And weβve solved the equation for
π₯ and π¦. π₯ equals negative one and π¦
equals three.

In fact, itβs always sensible to
check our answers by substituting them back into the equation and making sure that
it makes sense. If we do, we get five multiplied by
negative one plus two plus three multiplied by three minus five π. This does indeed give us negative
three plus four π as required.