Question Video: Solving Two-Variable Linear Equations with Complex Coefficients Mathematics

Video Transcript

Determine the real numbers 𝑥 and 𝑦 that satisfy the equation five 𝑥 plus two plus three 𝑦 minus five 𝑖 equals negative three plus four 𝑖.

Let’s look carefully at what we’ve been given. We have been given two complex numbers that we’re told are equal to each other. Now I know it doesn’t 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 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. And we’re told that 𝑥 and 𝑦 are real numbers. And this means that the expression five 𝑥 plus two must be real and three 𝑦 minus five must be real. So five 𝑥 plus two plus three 𝑦 minus five 𝑖 is a complex number. It has a real part of five 𝑥 plus two and an imaginary part of three 𝑦 minus five.

Next, we’ll recall what it actually means for two complex numbers to be equal. We see that two complex numbers 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 are equal if 𝑎 is equal to 𝑐 and 𝑏 is equal to 𝑑. In other words, their real parts must be equal and their imaginary parts must separately be equal.

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

Let’s repeat this process for the imaginary parts. We said that the imaginary part for our number on the left is three 𝑦 minus five. And on the right, we can see it’s four. This means that three 𝑦 minus five must be equal to four. We can add five to both sides of this equation. And then we’ll divide through by three. And we see that 𝑦 must be equal to three. And we’ve solved the equation for 𝑥 and 𝑦. 𝑥 equals negative one and 𝑦 equals three.

In fact, it’s 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 𝑖. This does indeed give us negative three plus four 𝑖 as required.