Daryl M. answered • 15d

Daryl's Math, Computers and Physics

As far as algebra is concerned, imaginary numbers can be worked with pretty much just like real numbers. All the usual rules of algebra apply.

So we have an equation:

14x - 64i = 42 + yi

To get this in a more standard form, we want to write y as a function of x. So we do the following operations:

- Flip the equality to get: 42 + yi = 14x - 64i
- Subtract 42 from both sides to get: y i = 14 x -64 i - 42
- Divide both sides by i to get: y = 14 x/i - 64 - 42/i
- Here's where a fact about i comes in: Since i * i = -1, we can write i = -1/i. Multiplying both sides by -1 gives: -i = 1/i. So we can replace 1/i by -i. Dividing by i is the same as multiplying by 1/i, which is the same as multiplying by -i. So the equation in 3 can be rewritten as: y = -14 x i - 64 + 42 i.

So the solutions to the equation are found by any pair of numbers (x,y) where y = -14 x i - 64 + 42 i. For example, we can let x=0 to find that one solution is y = -64 + 42 i. So the pair(0, -64 + 42 i) is one solution. We can check by plugging those into the original equation:

14x - 64i = 42 + yi

becomes

14*0 - 64i = 42 + (-64 + 42 i) * i

Simplifying gives:

-64i = 42 - 64 i + 42 i * i

Since i*i = -1, this simplifies further to:

-64i = 42 - 64i - 42

which is true.