MBAD 51415142 Tom Potter 7046852802 jtpotteruncc.edu

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MBAD 5141/5142Tom Potter704-685-2802jtpotter_at_un
cc.edu

• Class 1
• Lines

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Outline for todays class

• Introductions
• CENTRA Practice
• Syllabus/Course objectives
• Requirements
• Linear Functions
• Systems of Equations (Section 4.4)
• Applications to Business and Economics (Section
  4.5)
• Linear Inequalities (Section 11.1)

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Who is your instructor?

• Tom Potter
• Gaston County
• BS NC State University
• MA UNCC
• Phd UNCC (under construction!)
• Spent 11 years in industry.
• Teach math full time

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Markup Tools

• Toolbar Icons
• Yes/No
• Smiley Face
• When you want to say something!
• Need to step out?

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Now, you practice!
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Who Are You?
Name? Place of birth? Schools attended? Anything
else? Be sure to press CTRL when speaking
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Syllabus and Course Objectives

• Lets look at the syllabus
• Copies of syllabi are located on Blackboard
• You can get into Blackboard through 49er express.

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Cartesian Coordinates
Thanks to Renee Descartes for combining algebra
and geometry!
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The Cartesian coordinate system allows us to take
two variables and illustrate in visual form the
relationship between the two. To understand this
we need to review some basic definitions.
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The Cartesian Coordinate plane

• Is a grid of squares that is used as a standard
  setting to plot points.
• Each plane has two coordinate axes. The x-axis is
  horizontal and the y-axis is vertical. The two
  are perpendicular to each other.
• The intersection between the two axes is called
  the origin and is always given the coordinates
  (0,0)

y
(0,0)
x
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Cartesian Coordinates

• The relationship between two variables can be
  represented both in equation format and also
  graphically.
• Graphs are constructed using Cartesian
  Coordinates. Every point P(x,y) consists of a
  unique x-coordinate (abscissa) and y-coordinate
  (ordinate).

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Y
I (x,y)
II (-x,y)
2
4
2
X
III (-x,-y)
IV (x,-y)
-5
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In which quadrant are the following points
located?

• (-1,-5)
• Quadrant III
• (4,-2)
• Quadrant IV
• (-3,4)
• Quadrant II
• (0,2)
• Y-axis (HA! Fooled You!!!)
• (13602,0)
• X-axis

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Distance Formula

• The distance between two points can always be
  determined if the coordinates of each are known.
  The formula is below

The xs and ys come from the coordinates.
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Find the distance between the following points.

• P(3,5), Q(-2,4)
• 5.099
• P(3,0), Q(6,0)
• 3
• P(-4,-2), Q(-3,5)
• 7.071
• P(0,3), Q(0,17)
• 14

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Function Graphing

• The graph of an equation involving two variables,
  such as x and y, is the set (i.e., collection) of
  all points whose coordinates (x,y) satisfy the
  equation.
• When graphing it is sometimes helpful to create a
  table of values and create several points. Then
  draw a line though them to form the graph.

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For example, consider the equation
2x y - 3 0.
So, If I substitute the ordered pair (-2,-7) into
the equation, then I obtain 2(-2) (-7) 3 0.
I can do the same thing for (-1,-5), (0,-3) and
so on and always obtain 0 as my answer.
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Linear Equations

• Always graph a straight line
• Are determined by their steepness (slope) and
  their intercepts (points where they cross an
  axis)
• Are seen in the following forms

Standard Form
Slope-Intercept Form
Point-Slope Form
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Slope

• Is a measure of how steep a line is
• Is always determined by the relationship between
  the rise (distance in y-coordinate values) and
  the run (distance between x-coordinate values).
• Is always the ratio of rise to run.
• Is denoted by the letter m
• Can be determined by using the following formula

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Rules for slope

• Horizontal lines always have a slope of 0
  because they consist of points with the same
  y-coordinate. Example (3,2), (-7,2)
• Vertical lines always have an undefined slope or
  no slope. This is not the same as having a
  slope of 0. Example (4,7), (4, -12)
• All other lines have a real slope that is not
  0.

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Find the equation of a line that passes through
the points (2,5) and (6,-3)
(2,5)
m -2 (can you tell why?)
(6,-3)
So, slope is -2 and y-intercept is 9
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X- and Y- Intercepts

• The x-intercept is the point where the line
  crosses the x-axis. Since all points on the
  x-axis have a y-coordinate of 0 it is easy to
  determine the x-intercept by substituting 0 in
  for the y-coordinate and solve for x.
• To find the y-intercept (i.e., the point where
  the line crosses the y-axis) substitute 0 in
  for x and solve for y.

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Parallel and Perpendicular lines

• Lines that are parallel to each other have the
  same slopes.
• Lines that are perpendicular to each other have
  slopes whose product is -1.

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Parallel and Perpendicular lines Example 1

• Find the equation of the line that passes through
  the point (2,5) and is parallel to the line 3x -
  4y 3

m ¾
First, find the slope of the current line.
Since parallel lines have the same slope there is
no need to look further for a new slope. We can
now derive the new equation
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Parallel and Perpendicular lines Example 2

• Find the equation of the line that passes through
  the point (-3,2) and is perpendicular to the line
  y -3x 1

First, find the slope of the current line.
Next, find the slope of the new line.
Once the slope is found, derive the equation
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Applications of Linear Equations

• When producing a product, there are two types of
  costs involved. Fixed costs are those costs that
  are always the same, no matter how many units are
  produced. Variable costs depend on the level of
  production. It can, therefore, be determined
  that
• Total Cost Variable Cost Fixed Costs
• This is a form of a linear equation where the
  variable costs act as a slope and the fixed costs
  act as a y-intercept.

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The cost of manufacturing 10 copiers a day is
350, while it costs 600 to produce 20 copiers
per day. Assuming a linear cost relationship,
find a model that shows the total cost of
producing x copiers in a day. Then draw its
graph.
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Straight Line Depreciation

• Straight line depreciation reduces the initial
  cost of an asset by the same amount each year.
  That amount is a rate so it can be interpreted as
  a slope. The rate of depreciation (per year) is
  determined by the following formula
• D (Initial value Scrap value) (lifetime in
  Years)
• Now look at example 3 page 26

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Supply and Demand

• It is an economic law that the quantity x of any
  commodity that will be purchased by consumers
  depends on the price p at which that commodity is
  made available. In its simplest form, the Law of
  Demand is a linear relationship shown in the
  following equation
• p mx b
• A graph of a demand function is called the Demand
  Curve

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Supply and Demand (continued)

• The amount of a particular commodity x that
  suppliers are willing to make available also
  depends on the price p at which they can sell it.
  In its simplest form, the Law of Supply follows
  the same model as the Law of Demand
• p mx b
• A graph of a supply function is called the Supply
  Curve

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Supply and Demand (continued)

• It is interesting to not that demand curves have
  positive slopes and supply curves have negative
  slopes. Look on page 27 for examples.

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Systems of Equations

• A system of linear equations in two variables x
  and y consists of two linear equations in which
  the possible solution consists of one ordered
  pair that satisfies both equations. An example is
  given below

The solution is (4,2)
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Systems of Equations (continued)

• Solutions to linear systems can be determined
  algebraically. To do this there are two methods
  to choose from
• Substitution solve one equation for a variable
  and substitute that answer into the other
  equation and solve.
• Addition add the two equations together with
  the intent of eliminating one variable and
  solving for the remaining one. This often
  requires some manipulating through
  multiplication.

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Substitution Example

• Solve the following system

Step 1, solve for a variable
Step 3, find other variable solution
Step 2, substitute
So, solution is (-1,-2)
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Addition Example

• Solve the following system

Step 1, multiply both systems so that
cancellation can occur
Step 2, add the two together
Step 3, find other variable
So, solution is (2,1)
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Systems of Equations (continued)

• Solutions to linear systems can also be found
  graphically.
• If there is one unique solution, then the two
  functions will intersect at one point.
• If there are no solutions, the two functions will
  never intersect. In fact, they may be parallel to
  each other.
• There is the possibility that both functions
  actually graph the same line. In that case all
  real numbers are solutions.

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Systems of Equations (continued)

• Solve the following system graphically

(1,2)
So, the solution is the intersection at the point
(1,2)
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Helpful Hints to eliminate extra work

• If both equations in the system are linear, find
  the slope of each. If they are the same then
  solving for one solution is impossible.
• If, when solving algebraically, both variables
  are eliminated, then one solution is an
  impossibility.
• Sometimes its easier to graph than to solve.

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Applications to Business and Economics

• The break-even point is where the total cost of
  producing a product is the same as the total
  revenue from the sale of that product.
  Algebraically, it is where the cost function and
  revenue function intersect. See Figure 26 on page
  42 of your textbook for an example.

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Example of break even

• For a watchmaker, the cost of labor and materials
  per watch is 15 and fixed costs are 2000 per
  day. If each watch sells for 20, how many
  watches should be produced and sold each day to
  guarantee that the business breaks even?

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Example of break even (cont.)
Break-even point (400,8000)
12000
8000
Break even is where
4000
200
400
600
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Market Equilibrium

• Market equilibrium is obtained when the supply of
  a product meets the demand needs. It occurs at a
  price where the quantity supplied is equal to the
  quantity demanded.

p
supply
Market equilibrium (x,p)
demand
0
x
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Market Equilibrium Example

• If the demand and supply equations are

Determine the values of x and p at market
equilibrium
Therefore, the market equilibrium occurs at a
quantity (x) of 2 units of product at a price (p)
of 4 units.
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Linear Inequalities

• Linear inequalities are like linear equalities in
  that they consist of two variables (x,y). The
  difference between the two types is that there
  are many possible solutions instead of only one.
  Therefore it is necessary, when graphing, to
  include shaded areas to show solutions.

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Linear inequality example

• Graph the following linear inequality
• ygt2x - 4

Where do all of the solutions lie? Above or Below
the graph? To the right or left of it?
6
4
2
2
4
6
-2
-4
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Linear Inequalities continued

• When graphing a system of linear inequalities,
  the intersection of their solution fields is
  important. This type of problem comes up when we
  are interested in a range of values as our
  solution, rather than one unique x or y.

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Linear Inequalities continued

• Graph the following set of inequalities
• x0, y0, 3x2y 6, x-y 1

• x-y 1

3x2y 6
3
2
X0
1
1
2
3
-1
Y0