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Linear Equations

Today’s Homework Assignment
 
Section Problems
Outline

  Complex Numbers
Definitions
Complex Addition and Multiplication
Complex Division
Extending the Square Root

  Linear Equations
Definition
Manipulating Linear Equations
Problem Solving

  More Equations
Variable Denominators
Ratio and Proportion
More Problem Solving

Definitions

Definitions

Definition (The Imaginary Unit)
The imaginary unit i is a number such that i2 = −1.
Definition (Complex Numbers)
A complex number is any number that can be written in the form

a + bi,

where a and b are real numbers.

A complex number written as a +bi is said to be in standard form.
a is called the real part of the complex number
b is called the imaginary part of the complex number

Definitions

Examples of Complex Numbers

3 + 5i Already in standard form
10 − 4i 10 + (−4)i
7 7 + 0i
3i 0 + 3i

All real numbers are

Complex Addition and Multiplication

Complex Addition and Multiplication

To add complex numbers, just add the real parts and the imaginary
parts separately.

Definition
(a + bi) + (c + di) = (a + c) + (b + d)i

To multiply complex numbers, treat them like binomials and FOIL
the product.

Definition
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Complex Division

The Conjugate

Recall the special pattern

(a + b)(a − b) = a2 − b2.

Apply this to complex numbers (replace b with bi):

(a + bi)(a − bi) = a2 − (bi)2 = a2 − b2i2 = a2 + b2.

Changing the sign of the imaginary part is a useful operation: it
can be used to produce a real number!

Definition
The conjugate of a + bi is a − bi.

Note that a + bi is also the conjugate of a − bi.

Complex Division

Dividing by Complex Numbers

To divide by a complex number—that is, when a fraction has a
complex denominator—use the conjugate to transform the
denominator to a real number.

Examples

Examples

Extending the Square Root

A Better Definition

Our definition of i (”a number such that i2 = −1”) doesn’t really
define i. It just states a property of i .

We can make the definition more specific by stating that i will be
the principal square root of −1:
 

Definition
The principal square root of −b is denoted by and defined
as


Now we can define square roots of negative numbers :

Extending the Square Root

Complications

Recall the Properties of Radicals:

Properties of Radicals
If and are real numbers, then

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

Definition
A linear equation in the variable x is an equation that can be
written in the form ax + b = 0.

Examples

Manipulating Linear Equations

Solving Linear Equations

We may add, subtract , multiply, or divide by the same quantity on
both sides of an equation. The goal is to isolate the variable by
moving everything else to the other side.

It might be necessary to simplify first .

The Most Important Step

Always check your answers!

Examples Part 1

Examples

Problem Solving

Problem-Solving Example

Find three consecutive integers whose sum is 21.

We need to represent the three integers with variables. Call the
smallest one n. Then the other two are n + 1 and n + 2.
They add up to 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
if five times the smaller number is 11 less than four times the
larger number. 9 and 14

The average of the salaries of Kelly, Renee, and Nina is
$20,000 a year. If Kelly earns $4000 less than Renee, and
Nine’s salary is two-thirds of Renee’s salary, find the salary of
each person. $24,000 for Renee, $20,000 for Kelly,
$16,000 for Nina

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!


Oops!

This equation has no solution !

Examples Part 1

Examples

Ratio and Proportion
Cross-Multiplication

If both sides of an equation are single fractions, use
cross-multiplication:
The Cross-Multiplication Property
If b ≠0, d ≠ 0, and , then ad = bc.

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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