Linear Equations
Today’s Homework Assignment

Outline
Complex Numbers 

Definitions Definitions
A complex number written as a +bi is said to be in
standard form. 
Definitions Examples of Complex Numbers
3 + 5i Already in standard form 

Complex Addition and Multiplication Complex Addition and Multiplication
To add complex numbers, just add the real parts and
the imaginary
To multiply complex numbers, treat them like
binomials and FOIL

Complex Division The Conjugate Recall
the special pattern
Note that a + bi is also the conjugate of a − bi. 

Complex Division Dividing by Complex Numbers

Examples Examples


Extending the Square Root
A Better Definition

Extending the Square Root
Complications
These do not apply if b or c is negative! 

Extending the Square Root For instance, using the definition of we have
But, if we tried to use the Properties, we’d get which is wrong. 
Definition Linear Equations
Examples 

Manipulating Linear Equations
Solving Linear Equations
It might be necessary to simplify first . 
The Most Important Step Always check your answers! 

Examples Part 1 Examples 
Problem Solving ProblemSolving Example
Find three
consecutive integers whose sum is 21. Make sure you answer the question! The three integers are 6, 7, and 8. 

Examples Part 2 Examples
One number is 5 less than
another number. Find the numbers 
Variable Denominators Beware of Zero Denominators ! If a variable appears in a denominator, multiply by a common denominator to clear fractions, and take special care to ensure that the denominator is not zero. In other words, check your answers!
This equation has no solution! 

Examples Part 1 Examples 
Ratio and Proportion CrossMultiplication If both sides of an equation are single fractions, use crossmultiplication:


Examples Part 2 Examples 
More Problem Solving Examples One number is 100 larger than another number. If the larger number is divided by the smaller, the quotient is 15 and the remainder is 2. Find the numbers. 7 and 107 A sum of $2250 is to be divided between two people in the ratio of 2 to 3. How much does each person receive? $900 and $1350 Derek has some nickels and dimes worth $3.60. The number of dimes is one more than twice the number of nickels. How many nickels and dimes does he have? 14 nickels and 29 dimes The length of a rectangle is 2 inches less than three times its width. If the perimeter of the rectangle is 108 inches, find its length and width. 14 inches by 40 inches 
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