Post on 21-Dec-2015
MTH 070Elementary Algebra
Section 3.1Linear Equationsin Two Variables
Chapter 3Linear Equations, Slope, Inequalities,
And Introduction to Functions
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-LinearWhy?
2 3x y 4y x
6 2 3x y
1 2xy x 23 13yx
2 3 2y x x
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. Pick a value for one variable.2. Substitute the value into the equation.3. Solve for the second variable.4. Check (important !!)5. Give the solution as an ordered pair.
Option … create a table of solutions.
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 2006The College Board®
Assumes a 5% increase in college costs each year and a child entering college at age 18.
Source: FinAid Web Site09/14/07
http://www.finaid.org/savings/tuition-inflation.phtml
17-Year Trailing Averages
17-Year SpanCollege Inflation
General Inflation Rate Ratio
59-75 5.91% 3.79% 1.56
60-76 6.15% 4.07% 1.51
61-77 6.34% 4.38% 1.45
62-78 6.45% 4.76% 1.36
63-79 6.70% 5.37% 1.25
64-80 7.01% 6.05% 1.16
65-81 7.56% 6.62% 1.14
66-82 8.10% 6.89% 1.18
67-83 8.32% 6.87% 1.21
68-84 8.58% 6.95% 1.23
69-85 8.77% 6.91% 1.27
70-86 8.69% 6.68% 1.30
71-87 8.64% 6.56% 1.32
72-88 8.60% 6.55% 1.31
73-89 8.75% 6.66% 1.31
74-90 9.00% 6.61% 1.36
17-Year Trailing Averages
17-Year SpanCollege Inflation
General Inflation Rate Ratio
75-91 9.09% 6.19% 1.47
76-92 9.01% 5.81% 1.55
77-93 8.82% 5.66% 1.56
78-94 8.66% 5.42% 1.60
79-95 8.54% 5.13% 1.66
80-96 8.31% 4.64% 1.79
81-97 7.90% 4.00% 1.98
82-98 7.39% 3.46% 2.14
83-99 6.82% 3.21% 2.12
84-00 6.55% 3.26% 2.01
85-01 6.40% 3.19% 2.01
86-02 6.26% 3.07% 2.04
87-03 6.14% 3.11% 1.98
88-04 6.06% 3.04% 1.99
89-05 5.94% 2.99% 1.99
90-06 5.78% 2.89% 2.00
Source: FinAid Web Site09/14/07
http://www.finaid.org/savings/tuition-inflation.phtml
Breast Cancer Mortality Rate
1980 - 2005
Two problems with this graph?
Ordered Pairs
Two related values given in the form …
(x, y)x – Independent Variabley – Dependent VariableExample – Buying Gasoline
Independent Variable = # gallonsDependent Variable = cost(10, 27.90) 10 gallons of gas for $27.90
Rectangular Coordinate Systemaka: Cartesian Coordinate System or xy-plane
Descartes – Philosopher (“I think, therefore I am.”) & Mathematician (Analytic Geometry) – 1596-1650
Two perpendicular number lines.
x-axis
y-axis
origin
Rectangular Coordinate Systemaka: Cartesian Coordinate System or xy-plane
Every ordered pair of real numbers corresponds to one and only one point.Ordered Pair
Coordinates
Rectangular Coordinate Systemaka: Cartesian Coordinate System or xy-plane
Possible Locations of Points
Origin: (0,0)
QuadrantsI – NE: (+,+)II – NW: (–,+)III – SW: (–, –)IV – SE: (+, –)
Axes+ x axis: (+,0) right
– x axis: (–,0) left
+ y axis: (0,+) up
– y axis: (0, –) down
II I
III IV
Graphing:Linear Equations w/ 2 Variables
Always a line!2 points a line … so …1. Find 3 solutions.2. Plot the solutions.3. 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