From Section 1.4: Equations of Lines and Modeling Use a graphing calculator to model the data with a linear function. MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 127/68. L1: Study time (x); L2: Test grade (y). 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables 9.3 Matrices and Systems of Equations You may use the GeoGebra program to check your answers when solving a system of equations in two variables. Khan Academy is free and has terrific videos. I suggest you check it out. The algebra section is at http://www.khanacademy.org/#algebra TI calculator tutorial for this type problem http://cfcc.edu/faculty/cmoore/ti83modeling.htm Graphing Calculator Tutorial from textboo publisher http://media.pearsoncmg.com/aw/aw_mml_shared_1/gc_tutorial/start.html 9.1 Systems of Equations in Two Variables Solve a system of two linear equations in two variables by graphing. Solve a system of two linear equations in two variables using the substitution and the elimination methods. Use systems of two linear equations to solve applied problems. Systems of Equations A system of equations is composed of two or more equations considered simultaneously. Example: 5x y = 5 4x y = 3 This is a system of two linear equations in two variables. The solution set of this system consists of all ordered pairs that make both equations true. The ordered pair (2, 5) is a solution of this system. 1
Solving Systems of Equations Graphically When we graph a system of linear equations, each point at which the graphs intersect is a solution of both equations and therefore a solution of the system of equations. Solving Systems of Equations Graphically Let s solve the previous system graphically. 5x y = 5 4x y = 3 Solution: We see that the graph intersects at the single point (2, 5), so this is the solution of the system of equations. Systems of Equations If a system of equations has at least one solution, it is consistent. Illustration of Graphs Graphs of linear equations may be related to each other in one of three ways. If the system has no solutions, it is inconsistent. If a system of two linear equations in two variables has an infinite number of solutions, the equations are dependent. If a system of two linear equations in two variables represent two lines, they are independent. 2
Substitution Method The substitution method is a technique that gives accurate results when solving systems of equations. It is most often used when a variable is alone on one side of an equation or when it is easy to solve for a variable. One equation is used to express one variable in terms of the other, then it is substituted in the other equation. Example Use substitution to solve the system 5x y = 5, 4x y = 3. Solution Solve the first equation for y: y = 5x 5 Then we substitute 5x 5 for y in the second equation to give an equation in one variable. 4x (5x 5) = 3 4x 5x + 5 = 3 x = 2 Now we use back substitution and substitute 2 for x in either original equation. 4x y = 3 4(2) y = 3 8 y = 3 y = 5 We find the solution to the system of equations to be (2, 5), once again. Elimination Method Example Using the elimination method, we eliminate one variable by adding the two equations. If the coefficients of a variable are opposites, that variable can be eliminated by simply adding the original equations. If the coefficients are not opposites, it is necessary to multiply one or both equations by suitable constants, before we add. Solve the system using the elimination method. 6x + 2y = 4 10x + 7y = 8 Solution If we multiply the first equation by 5 and the second equation by 3, we will be able to eliminate the x variable. 30x + 10y = 20 Substituting: 6x + 2y = 4 30x 21y = 24 6x + 2( 4) = 4 11y = 44 6x 8 = 4 y = 4 6x = 12 x = 2 The solution is (2, 4). 3
Another Example Solve the system. x 3y = 9 (1) 2x 6y = 3 (2) Solution: 2x + 6y = 18 Mult. (1) by 2 2x 6y = 3 0 = 21 There are no values of x and y in which 0 = 21. So this system has no solution. The graphs of the equations are of parallel lines. Another Example Solve the system. 9x + 6y = 48 (1) 3x + 2y = 16 (2) Solution: 9x + 6y = 48 9x 6y = 48 Mult. (2) by 3 0 = 0 When we obtain the equation 0 = 0, we know the equations are dependent. There are infinitely many solutions. The graphs of the equations are identical. Application Ethan and Ian are twins. They have decided to save all of the money they earn, at their part time jobs, to buy a car to share at college. One week, Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256. The next week, Ethan worked 12 hours and Ian worked 16 hours and they earned $324. How much does each twin make per hour? Solution Letting E represent Ethan and I represent Ian, the following system can be obtained. 8E + 14I = 256 First week 12E + 16I = 324 Second week Mult by 12 96E + 168I = 3072 Mult by 8 96E 128I = 2592 40I = 480 I = 12 Solve for E. 8E + 14(12) = 256 8E = 88 E = 11 Ian makes $12 per hour while Ethan makes $11 per hour. 4
731/8. Solve graphically. x + y = 1 and 3x + y = 7 731/2. Match the system of equations with one of the graphs ( a) ( f), which follow. x y = 5 and x = 4y (1) x y = 5 (2) x = 4y You may use the GeoGebra program to check your answers when solving a system of equations in two variables. 731/24. Solve using the substitution method. Use a graphing calculator to check your answer. x 2y = 3 and 2x = 4y + 6 732/42. Solve using the elimination method. Also determine whether each system is consistent or inconsistent and whether the equations are dependent or independent. Use a graphing calculator to check your answer. 0.2x 0.3y = 0.3 and 0.4x + 0.6y = 0.2 5
732/51. Winter Sports Injuries. Skiers and snowboarders suffer about 288,400 injuries each winter, with skiing accounting for about 400 more injuries than snowboarding ( Source : U. S. Consumer Product Safety Commission). How many injuries occur in each winter sport? 733/56. Mail Order Business. A mail order lacrosse equipment business shipped 120 packages one day. Customers are charged $ 3.50 for each standard delivery package and $ 7.50 for each express delivery package. Total shipping charges for the day were $ 596. How many of each kind of package were shipped? 734/68. Motion. A DC10 travels 3000 km with a tail wind in 3 hr. It travels 3000 km with a head wind in 4 hr. Find the speed of the plane and the speed of the wind. 733/64. Break even Point. The point at which a company's costs equal its revenues is the break even point. In Exercises 61 64, C represents the production cost, in dollars, of x units of a product and R represents the revenue, in dollars, from the sale of x units. Find the number of units that must be produced and sold in order to break even. That is, find the value of x for which R = C. C = 3x + 400 and R = 7x 600 6