Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world problems. Eample : Gimos Model A Model B Total Available Dodads Sstem: The goal of solving a sstem is to find the values of and that satisf both equations simultaneousl. Three methods of solving a sstem we will investigate include graphing, solving algebraicall, and b using matrices. I. Graphical Solution When solving sstems of two linear equations, there are three possible outcomes:... In an attempt to solve the sstem in the calculator, we must first solve each equation for and then enter the equations in the calculator into Y and Y to find the intersection, if it eists. II. Algebraic Solution Elimination Method Eample: Substitution Method Eample: 9
III. Matrices An individual element in a matri is identified b its location in the matri. Coefficient matri Column vector Solution vector An identit matri is a matri that has the following characteristics: Back to Eample : 9 5 5 85 or AX Y Inverse Matri Method: Just as an inverse function undoes what a function does, the inverse matri undoes what the matri does. AX Y AA I A AX A Y X A Y We can get a solution to a sstem of equations b multipling the inverse of the coefficient matri b the (solution) column vector in the proper order (and order is critical!) on the graphing calculator. To set up the matrices on the calculator: Access the MATRIX command (over the ke) b using the kestrokes nd EDIT (enter the sie of the coefficient matri A b # rows #columns) ENTER (enter individual entries from the coefficient matri) ( nd quit). Then, MATRIX EDIT (arrow down to matri B) (enter sie of column vector) (enter individual entries from the column vector) ( nd quit). Now, to do the calculation, enter: MATRIX (select matri [A] if not automaticall selected) ENTER (times) MATRIX (select matri [B]) ENTER ENTER and read the result of the variable column vector. An augmented matri is the coefficient matri combined with the solution column vector. In order to solve a sstem in reduced row echelon form on the calculator, we must input the augmented matri. 5
Reduced Row Echelon Form Method: A matri is said to be in row echelon form provided all of the following conditions hold:... A matri is said to be in reduced row echelon form if both the following conditions hold:.. Calculator kestrokes for reduced row echelon form: MATRIX EDIT (enter augmented matri sie) ENTER (enter augmented matri individual entries) nd QUIT MATRIX MATH (scroll down to rref) ENTER MATRIX (select the augmented matri and close parentheses) ENTER and read the solution. Eample: Vitamin A Vitamin D Vitamin E Brand X Brand Y Brand Z Total Potential Solutions: There are three possibilities for solutions: A Consistent Sstem with One Solution: How can I tell? Geometric Interpretation: A Consistent Solution with Infinite Solutions: How can I tell? Geometric Interpretation: An Inconsistent Sstem: How can I tell? Geometric Interpretation: 5
Sstems of Equations and Matrices Activit Objectives: Solve sstem using substitution, elimination and graphing Solve sstem using matrices Solve sstem using matrices Set up sstem of equations and solve for applications 5
Solving Sstems. Each person in our group should use one of the following methods. Make sure ou all get the same result. Solve the following sstem of equations: a. Using elimination b. Using substitution c. Graphicall d. Using matrices 5. Solve the following sstem of equations: a. Using elimination b. Using substitution c. Graphicall d. Using matrices 5 5
Solving a Sstem Using an Inverse Matri: Eample: Consider: 8 5 8 Create matri: [A] to be and [B] to be. [A] - [B]= 5. Find the inverse matri using our calculator and them perform the multiplication to get the solution.. Now tr the method on the following sstem: 8 5
Applications Involving Sstems. A compan develops two different tpes of snack mi. Tpe A requires ounces of peanuts and 9 ounces of che mi while tpe B requires 5 ounces of peanuts and ounces of che mi. There is a total of 5 ounces of peanuts available and 85 ounces of che mi. How much of each tpe can the compan produce? Set up a sstem for the problem and solve it b an method ou like.. A person flies from Phoeni to Tucson and back (about miles each wa). He finds that it takes him hours to fl to Tucson against a headwind and onl hour to fl back with the wind. What was the airspeed of the plane (speed if there was no wind) and the speed of the wind? 55
. Two numbers when added together are 96 and when subtracted are 9. Find the two numbers. Set up a sstem for the situation and solve it b an method ou like.. A man has 9 coins in his pocket, all of which are dimes and quarters. If the total value of his change is $.55, how man dimes and how man quarters does he have? 56
57 Solving Sstems. Solve the following sstems using our graphing calculator. Determine if the sstem is consistent (one solution or infinitel man solutions) or inconsistent (no solution). If it has infinitel man solutions, write the dependent variable(s) in terms of the independent variable(s). a. 6 b. 6 7 8 c.
58 d. 7 5 e. 9 5 5 f. g. 8 5 5 6
Finding a Polnomial Given Points Eample: Set up a sstem of equations to find a parabola that passes through (,96), (, 9), and (5, 96). For the first point we know that a b c becomes 96 a b c so a b c 96 For the second point we know that a b c becomes 9 a b c so a b c 9 For the second point we know that a b c becomes 96 a 5 b5 c so 5a 5b c 96 So the sstem becomes: a b c 96 a b c 9 5a 5b c 96 5 5 96 9 96 6 8 96 So our quadratic would be a b c. Find a parabola that passes through the points: (-,), (,5), and (-6,5). Find a cubic function, a b c d, that passes through the points: (-,), (,5), (,), and (-5,). Note: In this case ou will have equations with variables so set up our sstem and use our graphing calculator to solve. 59
Applications Involving Sstems. In a particular factor, skilled workers are paid $5 per hour, unskilled workers are paid $9 per hour and shipping clerks are paid $ per hour. Recentl the compan has received an increase in orders and will need to hire a total of 7 workers. The compan has budgeted a total of $88 per hour for these new hires. Due to union requirements, the must hire twice as man skilled emploees as unskilled. How man of each tpe of worker should the compan hire?. I have 7 coins (some are pennies, some are dimes, and some are quarters). I have twice as man pennies as dimes. I have $.97 in coins. How man of each tpe of coin do I have? 6
. You have $ to invest in stocks. Stock A is predicted to ield % per ear. Stock B is the safest and is predicted to ield 5% per ear. Stock C is risk but is epected to ield 7% per ear. You decide to spend twice as much on stock B than on stock C. You hope to make % each ear on our stock. Use a matri (and our calculator) to find the amount of each tpe of stock that ou should bu.. An inheritance of $9,6 is to be split among children. To pa back mone owed to the two oldest children, it is written that the oldest child gets $, more than the oungest and the middle child get $, more than the oungest. How much should each child get? 6
5. There are candidates to choose for president and 8, people are epected to vote. Candidate A is epected to receive twice as man votes as candidate C. Candidate C is epected to receive times the votes as candidate B. Predict how man votes each candidate will receive. 6. Mar invested $, in three separate investments. At the end of the ear the earned %, 5% and 6%, respectivel. She invested twice as much in the account earning 5% as she did in the one paing %. She made a total of $5 in interest during the ear. How much did she allocate to each investment? 6