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Denis Waitley

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

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Denis Waitley

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  1. 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.”

  2. Intermediate Algebra 098A • Introduction • To • Linear Equations

  3. Def: Equation • An equation is a statement that two algebraic expressionshave the same value.

  4. Def: Solution • Solution: A replacement for the variable that makes the equation true. • Root of the equation • Satisfies the Equation • Zero of the equation

  5. 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

  6. 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

  7. Linear function • A function of form • f(x) = ax + b where a and b are real numbers and a is not equal to zero.

  8. 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.

  9. 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

  10. 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

  11. 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.

  12. Example

  13. Example

  14. Example

  15. Def: Equivalent Equations • Equivalent equations are equations that have exactly the same solutions sets. • Examples: • 5 – 3x = 17 • -3x= 12 • x = -4

  16. 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.

  17. 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.

  18. 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

  19. 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

  20. 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.”

  21. Useful Calculator Programs • CIRCLE • CIRCUM • CONE • CYLINDER • PRISM • PYRAMID • TRAPEZOI • APPS-AreaForm

  22. Robert Schuller – religious leader • “Spectacular achievement is always preceded by spectacular preparation.”

  23. Problem Solving • 1. Understand the Problem • 2. Devise a Plan • Use Definition statements • 3. Carry out a Plan • 4. Look Back • Check units

  24. 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.”

  25. Albert Einstein • “In the middle of difficulty lies opportunity.”

  26. 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.

  27. 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.

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