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
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