# The Math Daily News

### Imaginary Numbers & Solving Systems with 3 Variables

## Imaginary Numbers!? Don't worry, it's simple!

## EXAMPLES

## Example 1: Simplifying Basic Expressions

## Example 2: Adding and Subtracting

## Example 3: Multiplying

## Example 4: Simplifying Using Complex Conjugates

**Complex Conjugates**are two complex numbers in the form a+bi and a-bi. The product of complex conjugates is ALWAYS a real number, so they can be used to eliminate imaginary numbers from the denominator when simplifying an expression that has a complex number in the denominator.

In the work shown below, the numerator and denominator of the expression we are told to simplify is multiplied by the complex conjugate of the original denominator. In step 2, the product of those two expressions is found. In step 3, -1 is substituted for i^2, and as you can see, when step 4 simplified the expression, there is now a real number in the denominator. Finally, in step 5, everything is divided by 3, and the expressions is put into a+bi form, yielding a final answer of (4/15) + (2/15i).

## Recap: Solving Systems of Equations with 3 Variables

1) Eliminate one variable by using two pairs of equations.

2) Solve the system of equations to the two variables that weren't eliminated.

3) Substitute the two known variables into one of the original equations in order to find the third variable that was eliminated in step one.

4) Check your work! Substitute your 3 variables in to one of the original equations to make sure it works.

Mistakes in video:

- 2:27- I say 26, where it should say, as it is written on the whiteboard,
**-26** - 4:57 & 5:06- I say "write it in ordered pair form, whereas using the correct terminology, I should say
**"written as an ordered triple".**

## SUMMARY Of Concepts Learned

CONCEPT: Imaginary Numbers

-Simplify basic expressions

-Add and subtract

-Multiply

-Simplify using complex conjugates

CONCEPT: Solving Systems With 3 Variables

-Video of example

-Steps written out

## Real life application of systems of equations with 3 variables

**ordered triple**for a system of equations with 3 variables often represents points on a graph. That ordered triple is then directly involved in graphing planes in a three-dimensional space. Architects, surveyors, and cartographers use coordinate graphing in 3D everyday at work.