Matrices and Systems of Equations

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1 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 6: Systems of Equations and Matrices Section 6.3 Matrices and Systems of Equations Matrices A rectangular array of numbers is called a matrix (plural, matrices). The rows of a matrix are. The columns of a matrix are vertical. A matrix with m rows and n columns is said to be of order. Example 1 Determine the order of the following matrix: Augmented Matrix An augmented matrix is written using the coefficients and constants from a system of linear equations. Example 2 Write the augmented matrix for the following system of equations. -x 8y + 5z = 17-6x + 12z = -24 3x + y 8z = 11 Example 3 Write the system of equations that corresponds to the following augmented matrix. Example 4 Write the system of equations that corresponds to the following augmented matrix. Reduced Row-Echelon Form Our goal is take an augmented matrix from a system of linear equations and put it in reduced row-echelon form using Gauss-Jordan elimination.

2 Review of Solving Systems of Equations Using Substitution and/or Elimination 2 Example 5 Solve the system: 5x 3y = 14 4x + y = 18 Gauss-Jordan Elimination This is a method of solving systems of equations using matrices and is very useful if the systems are large. Steps for Gauss-Jordan Elimination and ROWOPS (calculator program) Write the augmented matrix on paper. Enter the matrix in the calculator: MATRX EDIT 1: Type in the order of the matrix and then the elements of the matrix, pressing ENTER after each input. The process is worked out column by column. Within each column you want to obtain 1 as the diagonal element first and then obtain the 0s within the rest of the column second. After a column is complete with 1 and then 0s, you move to the next column. To obtain 1 as the diagonal element, you multiply the entire row by the reciprocal of the current number. Write the row operation to be performed beside the paper matrix. On the calculator press PRGM and choose the program called ROWOPS. Press ENTER. You should see your original matrix displayed on the calculator screen. Double-check for data entry errors. Press ENTER again. You should see four choices. Choose 2 TO MULTIPLY and press ENTER. Enter in the row number and press ENTER and the reciprocal and press ENTER. Your next matrix will be displayed on the screen with a 1 as the first diagonal element. This 1 that was just obtained is called the pivot element. Write this matrix on paper. The next step is to obtain 0s in the rest of the column by using opposites. Multiply the row that you just obtained the 1 in by the opposite of the number to be changed and then add that result back to the row to be changed. Write this row operation beside the paper matrix. This process is called pivoting. Press ENTER in the calculator and then choose 3 TO PIVOT. You will need to enter the row number where the pivot is located (the 1 from above) and press ENTER and then the column number where the pivot is located and press ENTER. For this process the row and column numbers will always be the same because the pivot element is always on the diagonal. Your next matrix will be displayed on the screen with the column complete with 1 on the diagonal and 0s elsewhere. Write this matrix on paper. Move to the next column and go to the diagonal element and transform it into a 1 using the process from above. Then obtain 0s in the rest of the column. Always write your matrix on paper and write down the row operation to transition to the next matrix. When the matrix is in reduced-row echelon form, your answers will be the numbers in the last column.

3 Example 6 Use Gauss-Jordan elimination to solve the system: 5x 3y = 14 4x + y = 18 3 Example 7 Use Gauss-Jordan elimination to solve the system: -x + 2y + 6z = 2 3x + 2y + 6z = 6 x + 4y 3z = 1 Example 8 Use Gauss-Jordan elimination to solve the system: 3x + y + 3z = 1 x + 2y z = 2 2x y + 4z = 4 Example 9 Use Gauss-Jordan elimination to solve the system: x + y 3z = 4 4x + 5y + z = 1 2x + 3y + 7z = -7 Example 10 Solve the following problem using Gauss-Jordan elimination with matrices. Pat Summers wins $200,000 in the Louisiana state lottery. He invests part of the money in real estate with an annual return of 3% and another part in a money market account at 2.5% interest. He invests the rest, which amounts to $80,000 less than the sum of the other two parts, in certificates of deposit that pay 1.5%. If the total annual interest on the money is $4900, how much was invested at each rate?

4 Section 6.7 Systems of Inequalities & Linear Programming 4 Graphing a Linear Inequality Pretend the inequality symbol is an equals sign to graph the boundary line. If the inequality symbol is < or >, use a boundary line. If the inequality symbol is or, use a boundary line. Choose a test point NOT on the boundary line and plug it into the inequality. If the inequality is true, shade the side containing the test point. If the inequality is false, shade the other side. Example 1 Graph: y > x 4 Example 2 Graph: 4y 3x 5 Example 3 Graph: y < 2 Graphing Systems of Inequalities Graph each inequality on the same coordinate axes (same graph). Lightly shade or mark which side is to be shaded for each inequality. The intersection of the shaded regions is called the feasible region. This region is your solution set and needs to be marked by shading. The vertices ( points) are the points where the boundaries intersect that border the feasible region. The word vertices is plural for vertex. Example 4 Graph the system of inequalities. Then find the coordinates of the vertices. x -2 y < 3 2x Example 5 Graph the system of inequalities. Then find the coordinates of the vertices. 8x + 5y 40 x + 2y 8 x 0 y 0

5 Linear Programming Linear programming is used to optimize (maximize or minimize) values subject to certain constraints. For example, you may want to maximize profit but you are restricted in how many hours a day you can run a machine or the number of employees you can hire. Linear programming was developed during World War II. It was used to allocate supplies for the U.S. Air Force. Constraints are expressed as inequalities. The function that we want to maximize or minimize is called the function. 5 Fundamental Theorem of Linear Programming If an optimal value for a linear programming problem exists, it occurs at a vertex of the feasible region. Solving a Linear Programming Problem Graphically Identify the variables. Write the objective function including whether it is to be maximized or minimized. Write the constraints including the implied constraints. Graph the feasible region from the constraints and identify the vertices (corner points). Graph paper is very helpful for these problems. Substitute each vertex into the objective function. Based on whether the objective function is to be minimized or maximized, the smallest result would be the minimum and the largest result would be the maximum. Example 6 Find the maximum and minimum values of the function and the values of x and y for which they occur: Q = 28x 4y + 72 subject to: 5x + 4y 20 0 y 4 0 x 3 Example 7 An airline with two types of airplanes, P 1 and P 2, has contracted with a tour group to provide transportation for a minimum of 2000 first-class, 1500 touristclass, and 2400 economy-class passengers. For a certain trip, airplane P 1 costs $12,000 to operate and can accommodate 40 first-class, 40 tourist-class, and 120 economy-class passengers, whereas airplane P 2 costs $10,000 to operate and can accommodate 80 first-class, 30 tourist-class, and 40 economy-class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

6 Chapter 7: Conic Sections 6 Section 7.4 Nonlinear Systems of Equations & Inequalities Nonlinear Systems of Equations The solutions to these systems are points of intersection of the graphs from the system. You can expect to have no solution, one solution, or multiple solutions. We can solve nonlinear systems of equations by using the substitution or the elimination method. When some variables are squared and some are first-degree, try the substitution method. Solve one of the equations for one of the variables and substitute that result into the other equation. When all the variables have the same power, line them up vertically and try the elimination method. You may have to multiply one or both equations by a constant to be able to eliminate a variable. Example 1 Solve the system: x 2 + y 2 = 9 2x y = 3 Example 2 Solve the system: xy = 4 3x + 2y = -10 Example 3 Solve the system: 5x 2 2y 2 = -13 3x 2 + 4y 2 = 39 Example 4 For a student recreation building at Southport Community College, an architect wants to lay out a rectangular piece of land that has a perimeter of 204 m and an area of 2565 m 2. Find the dimensions of the piece of land. Nonlinear Systems of Inequalities These will be worked in the same manner as in section 5.7 except that at least one of the inequalities will not be linear. Example 5 Graph the solution set: x 2 + y 2 9 2x 3y > 6

6.3 Notes O Brien F15

6.3 Notes O Brien F15 CA th ed HL. Notes O Brien F. Solution of Linear Systems by ow Transformations I. Introduction II. In this section we will solve systems of first degree equations which have two or more variables. We will