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

Video Transcript

Find the real values of 𝑥 and 𝑦 that satisfy the equation 𝑥 plus 𝑦𝑖 equals 10 plus four 𝑖 squared.

If we get a complex number into the form 𝑎 plus 𝑏𝑖, 𝑎 and 𝑏 are real numbers. And for us, that means here both 𝑥 and 𝑦 are real numbers. We have to take 10 plus four 𝑖 squared and put it into the format of 𝑎 plus 𝑏𝑖.

At first glance, you might be tempted to take the square root of both sides. However, once we’re here, we end up with a whole host of other problems and we’re no closer to the format 𝑎 plus 𝑏𝑖. Instead of doing that, we want to actually square 10 plus four 𝑖.

We want to find out what 10 plus four 𝑖 times 10 plus four 𝑖 is. We can FOIL these two terms. 10 times 10 equals 100. 10 times four 𝑖 equals 40𝑖. Four 𝑖 times 10 equals 40𝑖. And four 𝑖 times four 𝑖 equals 16, four times four equals 16, 𝑖 squared, 𝑖 times 𝑖 equals 𝑖 squared.

We combine our like terms. And now, our equation says 𝑥 plus 𝑦𝑖 equals 100 plus 80𝑖 plus 16𝑖 squared. You have to remember that the imaginary number 𝑖 is equal to the square root of negative one. 𝑖 squared equals the square root of negative one squared, negative one. 𝑖 squared equals negative one.

In our equation, in place of 𝑖 squared, we substitute negative one. And then, we have 16 times negative one equals negative 16. 100 plus 80𝑖 minus 16. We can combine 100 and negative 16 to get 84 plus 80𝑖. Now, our equation is in the correct format 𝑥 plus 𝑦𝑖 equals 84 plus 80𝑖. 𝑥 equals 84 and 𝑦 equals 80.

The real values for 𝑥 and 𝑦 are 84 and 80.