Denis Waitley

• “Failure should be our teacher, not our undertaker. Failure is delay, not defeat. It is a temporary detour, not a dead end. Failure is something we can avoid only by saying nothing, doing nothing, and being nothing.”

Intermediate Algebra 098A

•Introduction

•To

•Linear Equations

Def: Equation

•An equation is a statement that two algebraic expressions

have the same value.

Def: Solution

• Solution: A replacement for the variable that makes the equation true.

• Root of the equation• Satisfies the Equation• Zero of the equation

Def: Solution Set

• A set containing all the solutions for the given equation.

• Could have one, two, or many elements.

• Could be the empty set

• Could be all Real numbers

Def: Linear Equation in One Variable

• An equation that can be written in the form ax + b = c where a,b,c are real numbers and a is not equal to zero

Linear function

• A function of form

• f(x) = ax + b where a and b are real numbers and a is not equal to zero.

Equation Solving: The Graphing Method

• 1. Graph the left side of the equation.

• 2. Graph the right side of the equation.

• 3. Trace to the point of intersection

• Can use the calculator for intersect

• The x coordinate of that point is the solution of the equation.

Equation solving - graphing

• The y coordinate is the value of both the left side and the right side of the original equation when x is replaced with the solution.

• Hint: An integer setting is useful

• Hint: x setting of [-9.4,9.4] also useful

Def: Identity

• An equation is an identity if every permissible replacement for the variable is a solution.

• The graphs of left and right sides coincide.

• The solution set is R

R

Def: Inconsistent equation

• An equation with no solution is an inconsistent equation.

• Also called a contradiction.

• The graphs of left and right sides never intersect.

• The solution set is the empty set.

Example

119 2 6

2x x

Example

3 1x x

Example

3 3x x

Def: Equivalent Equations

• Equivalent equations are equations that have exactly the same solutions sets.

• Examples:

• 5 – 3x = 17

• -3x= 12

• x = -4

Addition Property of Equality

• If a = b, then a + c = b + c

• For all real numbers a,b, and c.

• Equals plus equals are equal.

Multiplication Property of Equality

• If a = b, then ac = bc is true

• For all real numbers a,b, and c where c is not equal to 0.

• Equals times equals are equal.

Solving Linear Equations

• Simplify both sides of the equation as needed.– Distribute to Clear parentheses– Clear fractions by multiplying by the LCD– Clear decimals by multiplying by a power of 10

determined by the decimal number with the most places

– Combine like terms

Solving Linear Equations Cont:

• Use the addition property so that all variable terms are on one side of the equation and all constants are on the other side.

• Combine like terms.

• Use the multiplication property to isolate the variable

• Verify the solution

Ralph Waldo Emerson – American essayist, poet, and philosopher (1803-1882)

• “The world looks like a multiplication table or a mathematical equation, which, turn it how you will, balances itself.”

Useful Calculator Programs

• CIRCLE

• CIRCUM

• CONE

• CYLINDER

• PRISM

• PYRAMID

• TRAPEZOI

• APPS-AreaForm

Robert Schuller – religious leader

• “Spectacular achievement is always preceded by spectacular preparation.”

Problem Solving

• 1. Understand the Problem• 2. Devise a Plan

– Use Definition statements

• 3. Carry out a Plan• 4. Look Back

– Check units

Les Brown

• “If you view all the things that happen to you, both good and bad, as opportunities, then you operate out of a higher level of consciousness.”

• Albert Einstein

»“In the middle of difficulty lies opportunity.”

Intersection - Disjunction

• Intersection: For two sets A and B, the intersection of A and B, is a set containing only elements that are in both A and B.

A B

Union - conjunction

• For two sets A and B, the union of A and B is a set containing every element in A or in B.

A B