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