1. MTH 070 Elementary Algebra Section 3.1 Linear Equations in Two Variables Chapter 3 Linear Equations, Slope, Inequalities, And Introduction to Functions Copyright © 2010 by Ron Wallace, all rights reserved.

2. 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)

3. Linear or Non-Linear Why?

4. Solutions A pair of values for the variables that make the equation true. Express as an ordered pair: ( x, y ) Example: 2x + y = 10

5. 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.

6. 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

7. 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 …

8. Source: State Farm Insurance Web Site (09/14/07) http://www.statefarm.com/learning/life_stages/learning_lifestages_college.asp

9. 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.

10. Source: FinAid Web Site 09/14/07 http://www.finaid.org/savings/tuition-inflation.phtml 17-Year Trailing Averages 17-Year Span College Inflation General InflationRate Ratio 59-755.91%3.79%1.56 60-766.15%4.07%1.51 61-776.34%4.38%1.45 62-786.45%4.76%1.36 63-796.70%5.37%1.25 64-807.01%6.05%1.16 65-817.56%6.62%1.14 66-828.10%6.89%1.18 67-838.32%6.87%1.21 68-848.58%6.95%1.23 69-858.77%6.91%1.27 70-868.69%6.68%1.30 71-878.64%6.56%1.32 72-888.60%6.55%1.31 73-898.75%6.66%1.31 74-909.00%6.61%1.36 17-Year Trailing Averages 17-Year Span College Inflation General InflationRate Ratio 75-919.09%6.19%1.47 76-929.01%5.81%1.55 77-938.82%5.66%1.56 78-948.66%5.42%1.60 79-958.54%5.13%1.66 80-968.31%4.64%1.79 81-977.90%4.00%1.98 82-987.39%3.46%2.14 83-996.82%3.21%2.12 84-006.55%3.26%2.01 85-016.40%3.19%2.01 86-026.26%3.07%2.04 87-036.14%3.11%1.98 88-046.06%3.04%1.99 89-055.94%2.99%1.99 90-065.78%2.89%2.00

11. Source: FinAid Web Site 09/14/07 http://www.finaid.org/savings/tuition-inflation.phtml

12. Breast Cancer Mortality Rate 1980 - 2005 Two problems with this graph?

13. 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

14. 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. x-axis y-axis origin

15. Rectangular Coordinate System aka: Cartesian Coordinate System or xy-plane Every ordered pair of real numbers corresponds to one and only one point. Ordered Pair  Coordinates

16. 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: (+, –) Axes + x axis: (+,0) right – x axis: (–,0) left + y axis: (0,+) up – y axis: (0, –) down III IIIIV

17. 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.

18. Example 1 of 2 Graph: x + 2y = 6

19. Example 2 of 2 Graph: y = x + 2