Download MTH 060 Elementary Algebra I

MTH 070
Elementary Algebra
Chapter 3
Linear Equations, Slope, Inequalities,
And Introduction to Functions
Section 3.1
Linear Equations
in Two Variables
Copyright © 2010 by Ron Wallace, all rights reserved.
Linear Equations w/ 2 Variables
Equations
(once grouping symbols are removed
and like terms are combined) that
include
three terms: ax, by, and c (a, b, and c are
constants & c may be zero)
Standard Form:
Ax + By = C
Solved for Y:
y = mx + b
(A, B, C, m, & b are constants)
Linear or Non-Linear
Why?
x  2y  3
y  x4
3x  2y  13
xy  1  2 x
y  x  3x  2
6x  2 y  3
2
Solutions
A pair of values for the variables
that make the equation true.
Express as an ordered pair: (x, y)
Example:
2x + y = 10
Solutions
A pair of values for the variables
that make the equation true.
How many solutions?
countless
Strategy …
1.
2.
3.
4.
5.
Option … create
a table of
solutions.
Pick a value for one variable.
Substitute the value into the equation.
Solve for the second variable.
Check (important !!)
Give the solution as an ordered pair.
Solutions - Example
y = 4x – 1
Find the solution when
x=3
Find the solution when
y = –5
Find the solution when
x=0
Find the solution when
y=0
Graphing
Since all solutions cannot be listed,
all solutions can be expressed by a
picture – called a graph.
Equations with 2 variables describe
a relationship between two
quantities, as seen in the following
graphs …
Source: State Farm Insurance Web Site
(09/14/07)
http://www.statefarm.com/learning/life_stages/learning_lifestages_college.asp
Source: Trends in College Pricing 2006
The College Board®
Assumes a 5% increase in college costs each year and a child entering college at age 18.
Source: FinAid Web Site
09/14/07
http://www.finaid.org/savings/tuition-inflation.phtml
17-Year Trailing Averages
17-Year Trailing Averages
17-Year Span
College
Inflation
General
Inflation
17-Year Span
College
Inflation
General
Inflation
Rate Ratio
Rate Ratio
59-75
5.91%
3.79%
1.56
75-91
9.09%
6.19%
1.47
60-76
6.15%
4.07%
1.51
76-92
9.01%
5.81%
1.55
61-77
6.34%
4.38%
1.45
77-93
8.82%
5.66%
1.56
62-78
6.45%
4.76%
1.36
78-94
8.66%
5.42%
1.60
63-79
6.70%
5.37%
1.25
79-95
8.54%
5.13%
1.66
64-80
7.01%
6.05%
1.16
80-96
8.31%
4.64%
1.79
65-81
7.56%
6.62%
1.14
81-97
7.90%
4.00%
1.98
66-82
8.10%
6.89%
1.18
82-98
7.39%
3.46%
2.14
67-83
8.32%
6.87%
1.21
83-99
6.82%
3.21%
2.12
68-84
8.58%
6.95%
1.23
84-00
6.55%
3.26%
2.01
69-85
8.77%
6.91%
1.27
85-01
6.40%
3.19%
2.01
70-86
8.69%
6.68%
1.30
86-02
6.26%
3.07%
2.04
71-87
8.64%
6.56%
1.32
87-03
6.14%
3.11%
1.98
72-88
8.60%
6.55%
1.31
88-04
6.06%
3.04%
1.99
73-89
8.75%
6.66%
1.31
89-05
5.94%
2.99%
1.99
74-90
9.00%
6.61%
1.36
90-06
5.78%
2.89%
2.00
Source: FinAid Web Site
09/14/07
http://www.finaid.org/savings/tuition-inflation.phtml
Ratio of Tuition Inflation to General Inflation
17 Year Trailing Averages
2.50
2.00
1.50
1.00
0.50
0.00
Breast Cancer Mortality Rate
1980 - 2005
Two problems
with this graph?
Ordered Pairs
Two related values given in the form …
(x, y)
x – Independent Variable
y – Dependent Variable
Example – Buying Gasoline
Independent Variable = # gallons
Dependent Variable = cost
(10, 27.90)  10 gallons of gas for $27.90
Rectangular Coordinate System
aka: Cartesian Coordinate System or xy-plane
Descartes –
Philosopher (“I think, therefore I am.”)
& Mathematician (Analytic Geometry) – 1596-1650
Two perpendicular number lines.
y-axis
x-axis
origin
Rectangular Coordinate System
aka: Cartesian Coordinate System or xy-plane
Ordered Pair  Coordinates
Every ordered pair
of real numbers
corresponds to one
and only one point.
Rectangular Coordinate System
aka: Cartesian Coordinate System or xy-plane
Possible Locations of Points
Origin: (0,0)
Quadrants
I – NE: (+,+)
II – NW: (–,+)
III – SW: (–, –)
IV – SE: (+, –)
II
III
Axes
+ x axis: (+,0) right
– x axis: (–,0) left
+ y axis: (0,+) up
– y axis: (0, –) down
I
IV
Graphing:
Linear Equations w/ 2 Variables
Always a line!
2 points  a line … so …
1.
2.
3.
Find 3 solutions.
Plot the solutions.
Draw the line.
What does the graph represent?
Picture of ALL solutions.
Example 1 of 2
Graph:
x + 2y = 6
Example 2 of 2
Graph:
y=x+2