Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 x4 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