Chapter 11 Sstems of Equations 11.1 Solving Sstems of Linear Equations b Graphing Learning Objectives: A. Decide whether an ordered pair is a solution of a sstem of linear equations. B. Solve a sstem of linear equations b graphing. C. Without graphing, determine the number of solutions of a sstem. Ke Vocabular: sstem of linear equations, parallel lines, no solution, infinite number of solutions, inconsistent sstem, consistent sstem, dependent equation, independent equations. Definitions: A sstem of linear equations consists of two or more linear equations. Graphicall, a solution of a sstem of two equations in two variables is a point common to the graphs the intersection point(s) of the lines. Three tpes of the Sstem of Equations Consistent Sstem Eactl one solution: The equations are independent The two lines intersect at one point (,) The slopes of the lines are not equal The solution is (,), where and are two real numbers. Infinitel man solutions: The equations are dependent The graphs of the two lines are identical The slopes of the lines are equal, and the - intercepts are equal, too When we solve the sstem, we get a true statement An ordered pair (,) where and are real numbers is a solution to the sstem Inconsistent Sstem: has no solution the equations are independent the graphs of the two lines are parallel the slopes of the lines are equal, but the intercepts are not When we solve the sstem, we get a false statement There are no ordered pairs (,) that is a solution to the sstem 1
Class notes: Eample of sstem of equations: 1 1.. Solution of the sstem is the point(s) where the graphs intersect. Eample 1. Is ( ),9 a solution of the sstem? Eample. Solve, graph and state number of solution. 1.. 8 6
11.1 Eercises 1. Use the graph to find the solution of the sstem of linear equations. Is the sstem consistent or inconsistent? If it s consistent, are the equations dependent or independent? a. b... a. (,1) b. (,1) a. (, 6) b. ( 16, ) 19 6 1 a. ( 6,) b. (, ) c. Solve each sstem of equations b graphing. Check the solution. Determine whether the sstem is consistent or inconsistent, and if it is consistent whether the equations dependent or independent. Determine whether each ordered pair is a solution of the sstem of linear equations... 6. 7. 8. 6 8 1 8 1
9. 1 17. The sum of twice a number and another number is 9. Their difference is 1. 1. 6 1 1 11. 1. 6 1. Consider the linear equation 1. Find another linear equation such that the two equation form a sstem that is a. consistent with independent equations b. consistent with dependent equations c. inconsistent. 1. Consider the linear equation. Find another linear equation such that the two equations provide a. eactl one solution b. no solution c. infinitel man solutions For 1 17, find the unknown numbers b solving the sstem of equations ou set up for the problems. 1. The sum of two numbers is 6 and their difference is. 16. Half of a number added to another number equals 8. Their difference is 11. 18. A cell phone compan, Cheap Chat offers a plan with $9 monthl fee and 9 cents per minute. Another compan, Terrific Talk has a $ monthl fee and 11 cents per minute. The cost equations are: for Cheap Chat (Line a) C 9.9 for Terrific Talk (Line b) C.11 where is the number of minutes used. The graph of the equations is given: 1 C a. When is the Cheap Chat cheaper? b. When is the Terrific Talk cheaper? c. Interpret the meaning of the intersection point in this contet. 19. You want to join a health club, and there are two of those close to our house. The Muscle Maniac Club charges $ initiation fee, and $1 per month. The other Club, Un Couch Potato charges $1 initiation fee, and $ per month. a. Find the cost equations for both clubs. b. Graph the lines and find the point of intersection. b a 1
c. After how man months are the costs the same? d. If ou are planning to join the club onl for a ear, which club would be less epensive? e. If ou have long term plans, joining the club for more than two ears, which club would be less epensive?