1. SYSTEMS OF EQUATIONS MATH10 ALGEBRA Systems of Equations(Algebra and Trigonometry, Young 2nd Edition, page 874-904)

3. Graph systems of linear equations.

4. Understand that systems of linear equations may have one solution, no solution, or infinitely many solutions.

7. To solve systems of linear equations in two variables using the elimination method.

8. To solve systems of linear equations in two variables by graphing.

10. Week 6 Day 1 Example: y 6 3x-y=11 x+2y=6 Check x -4 INTERPRETATION

11. Week 6 Day 1 THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS y 6 1. Independent System One solution x -4 Lines have different slopes.

12. Week 6 Day 1 THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS y 6 2. Inconsistent System No solution x -4 Lines are parallel (same slopes and different y - intercepts.)

13. Week 6 Day 1 THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS y 6 3. Dependent System Infinitely many solution x -4 Lines coincide (same slopes and same y – intercepts).

15. Week 6 Day 1 DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS ONE SOLUTION

16. Week 6 Day 1 DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS NO SOLUTION

17. Week 6 Day 1 DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS INFINITE SOLUTION

18. Week 6 Day 1 ELIMINATION METHOD STEPS: Multiply the coefficients of one or both of the equations so that one of the variables will be eliminated when two equations are added. Eliminate one of the variables by adding the expression found in Step 1 to the other equation. The result is an equation in one variable. Solve the equation obtained in Step 2. Back substitute the value found in Step 3 into either of the original equation. Check that the solution satisfies both equations.

19. Week 6 Day 1 DETERMINING BY ELIMINATION THAT A SYSTEM HAS ONE SOLUTION

20. Week 6 Day 1 DETERMINING BY ELIMINATION THAT A SYSTEM HAS NO SOLUTION

21. Week 6 Day 1 DETERMINING BY ELIMINATION THAT A SYSTEM HAS INFINITELY MANY SOLUTION

22. Week 6 Day 1 GRAPHING METHOD STEPS: Write the equations in the slope-intercept form. Graph the lines. Identify the points of intersection. Check that the solution satisfies both equations. OR Solve for the x and y intercepts. Graph the lines. Identify the points of intersection. Check that the solution satisfies both equations.

23. Week 6 Day 1 DETERMINING BY GRAPH THAT A SYSTEM HAS ONE SOLUTION, NO SOLUTION OR INFINITELY MANY SOLUTION

25. Use graphing to confirm the solution(s) found using either elimination or substitution. EXAMPLE State which of the two algebraic methods (elimination or substitution) would be the preferred method to solve each system of linear equations.

28. To identify three types of solutions: one solution, no solution or infinitely many solutions.

30. The graph of any equation in three variables requires three dimensional coordinate system.

31. In two variables, the graph of a linear equation is a line, while in three variables the graph of a linear equation is a plane which can be thought of as an infinite sheet of paper.

33. Week 6 Day 2 One Solution Solution

34. Week 6 Day 2 No Solution or

35. Week 6 Day 2 Infinitely Many Solutions Solution (line of intersection)

36. Week 6 Day 2 SOLVING SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES USING ELIMINATION AND SUBSTITUTION STEPS Reduce the system of three equations in three variables to two equations in two (of the same) variables by applying elimination. Solve the resulting system of two linear equations in two variables by applying elimination or substitution. Substitute the solution in Step 2 into any of the original equations and solve for the third variable. 4. Check that the solution satisfies all three original equations.

37. Week 6 Day 2 EXAMPLE

39. Week 6 Day 2 No solution if the two planes are parallel z y x

40. Week 6 Day 2 Infinitely many solutions if the two planes intersect in a line. z solution y x

41. Week 6 Day 2 EXAMPLE

45. Week 6 Day 3 QUADRATIC SYSTEMS IN TWO VARIABLES 2. Two Quadratics 1. One Linear, One Quadratic 3.Two Quadratics, all terms containing the variable are of second degree College Algebra Revised edition, Catalina D. Mijares page 241-250

46. Week 6 Day 3 QUADRATIC SYSTEMS IN TWO VARIABLES 4. Symmetric Quadratic Equation 5.Other types which does not fall on the previous types. College Algebra Revised edition, Catalina D. Mijares page 241-250

47. Week 6 Day 3 APPLICATION INVOLVING SYSTEMS OF LINEAR EQUATIONS Week 6 Day 3 Application Involving Systems of Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 886-891 and 899-904).

48. Week 6 Day 3 RECALL Start A Read and analyze the problem Make a diagram or sketch if possible Solve the equation Determine the unknown quantity. Check the solution Set up an equation, assign variables to represent what you are asked to find. no Is the unknown solved? no yes yes Did you set up the equation? A End

49. Week 6 Day 3 APPLICATION 1. Upon graduation with a degree of management information systems(MIS), you decide to work for a company that buys data from the states’ department of motor vehicles and sells to banks and car dealerships customized reports detailing how many cars at each dealership are financed through particular banks. Autocount Corporation offers you a $15,000 base salary and a 10% commission on your total annual sales. Polk Corporation offers you a base salary of $30,000 plus a 5% commission on your total annual sales. How many total sales would you have to make per year to make more money at Autocount? (# 59 page 890)

50. Week 6 Day 3 APPLICATION 2. A mechanic has 340 gallons of gasoline and 10 gallons of oil to make gas/oil mixtures. He wants one mixture to be 4% oil and the other mixture to be 2.5% oil. If he wants to use all of the gas and oil, how many gallons of gas and oil are in each of the resulting mixtures? (# 58 page 890) 3. A direct flight on Delta Airlines from Atlanta to Paris is 4000 miles and takes approximately 8 hours going East (Atlanta to Paris) and 10 hours going West ( Paris to Atlanta). Although the plane averages the same airspeed, there is a headwind while traveling west and a tailwind while travelling east resulting in different airspeeds. What is the average airspeed of the plane and what is the average wind speed ? (# 63 page 890)

51. Week 6 Day 3 APPLICATION Suppose you’re going to eat only Subway sandwiches for a week (7 days) for lunch and dinner (total o0f 14 meals). 4. Your goal is a total of 4840 calories and 190 grams of fat. How many of each sandwich would you eat that week to obtain this goal? ( #33 page 901)

52. Week 6 Day 3 APPLICATION 5. Tara and Lamar decide to place $20,000 of their savings into investments. They put some in a money market account earning 3% interest, some in a mutual fund that has been averaging 7% a year, and some in a stock that rose 10% last year. If they put $6,000 more in the money market than in the mutual fund and the mutual fund and stocks have the same growth in the next year as they did in the previous year , they will earn $1,180 in a year. How much money did they put in each of the three investments? (# 39 page 902)