Transcript Intermediate – Sections 3.1-3.2

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
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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
1
x  19  2 x  6
2
Example
x  3  1 x
Example
x 3  3 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
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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