Pre-Calculus Notes: Chapter 2 Systems of Linear Equations and Inequalities

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1 Name: Pre-Calculus Notes: Chapter 2 Systems of Linear Equations and Inequalities Section 1 Solving Systems of Equations in Two Variables System of equations Solution to the system Consistent system Independent system Dependent system Inconsistent system Systems of equations can be solved using one of three different methods: Graphing Substitution Elimination Example 1 Solve the system of equations by graphing. y = 4x 18 3 y = x 5 4 1

2 Example 2 Use the substitution method to solve the system of equations. y = 3x 8 2x + y = 22 Example 3 Use the elimination method to solve the system of equations. 5x + 2y = 340 3x 4y = 360 Example 4 Madison is thinking about leasing a car for two years. The dealership says that they will lease her the car she has chosen for $326 per month with only $200 down. However, if she pays $1600 down, the lease payment drops to $226 per month. What is the break-even point when comparing these lease options? Which 2-year lease should she choose if the down payment is not a problem? 2

3 Section 2 Solving Systems of Equations in Three Variables Example 1 Solve the system of equations. 3y = -9z 4x + 2y 2z = 0-3x y + 4z = -2 Example 2 Solve the system of equations. 5x 2y + z = 11 2x + y + 3z = 0 6x 2y 2z = 16 3

4 Example 3 In the 1998 WNBA season, Sheryl Swoopes made 83% of her 86 attempted free throws. She made 244 of her 1-point, 2-point, and 3-point attempts, resulting in 453 points. Find the number of 1-point free throws, 2-point field goals, and 3-point field goals Swoopes made in the 1998 season. 4

5 Section 3 Modeling Real-World Data with Matrices Matrix m n matrix Dimensions 33matrix 21matrix 23matrix There are special names for certain matrices: Row Matrix Column Matrix Square Matrix nth order Matrix Equal Matrices Example 1 During the summer, Ms. Robbins received several types of grains on her farm to feed her livestock. Use a matrix to represent the data. June 15,000 bushels corn, 2000 bushels soybeans, 500 bushels oats July 13,500 bushels corn, 6500 bushels soybeans, 1000 bushels oats August 14,000 bushels corn, 5500 bushels soybeans, 1500 bushels oats 5

6 Exploring Matrix Operations - For each operation, determine the rule. Matrix Addition ) ) Rule: Zero Matrix Additive Inverse Matrix Subtraction ) ) Rule: Scalar Multiplication 5.) ) Rule: Matrix Multiplication ) ) ) ) ) Rule: 6

7 Example 2 Find the values of x and y for which the matrix equation y 4x y 3 2x 1 is true. Example 3 Find A + B if 7 4 A 5 0 and B Example 4 Find S T if S and T Example If A 3 8, find 4A

8 Example 6 Use matrices A 0 1 0, B, C to find each product a.) BC b.) CB c.) AC d.) CA Example 7 At Ohio State University, professional students pay different tuition rates based on the programs they have chosen. For the school year, in-state students in the school of medicine paid $5646 per quarter, dental school students paid $4792 per quarter, and veterinary medicine students paid $4405 per quarter. The chart lists the total student enrollment in those programs for each quarter of the school year. Use matrix multiplication to find the amount of tuition paid for each of these four quarters. Quarter Enrollment Med. Dent. Vet. Autumn Winter Spring Summer Source: The Ohio State University Registrar 8

9 Section 4 Modeling Motion with Matrices Transformations translations reflections rotations dilations Triangle ABC can be represented by the following vertex matrix. Triangle A B C is congruent to and has the same orientation as ABC, but is moved from ABC s location. The coordinates of A' B' C' can by expressed as the following vertex matrix: Compare the two matrices. If you add to the first matrix you get the second matrix. This type of matrix is called a. In this transformation ABC is the and A' B' C' is the after the translation. 9

10 Example 1 Suppose the quadrilateral RSTU with vertices R(3, 2), S(7, 4), T(9, 8), and U(5, 6) is translated 2 units right and 3 units down. a.) Represent the vertices of the quadrilateral as a matrix. b.) Write the translation matrix. c.) Use the translation matrix to find the vertices of R S T U, the translated image of the quadrilateral. d.) Graph the quadrilateral RSTU and its image. 10

11 Reflections over the x-axis Reflect the point (1,2) over the x-axis. How did the coordinate change? 1 What matrix could you multiply 2 by to yield your new coordinate? Reflections over the y-axis Reflect the point (1,2) over the y-axis. How did the coordinate change? 1 What matrix could you multiply 2 by to yield your new coordinate? Reflections over the line y = x Reflect the point (1,2) over the line y = x. How did the coordinate change? 1 What matrix could you multiply 2 by to yield your new coordinate? Reflection Matrices For a reflection over the: Symbolized by: Multiply the vertex matrix by: x-axis y-axis line y = x 11

12 Rotations about the origin of 90 o Rotate the point (1,2) about the origin How did the coordinate change? 1 What matrix could you multiply 2 by to yield your new coordinate? Rotations about the origin of 180 o Rotate the point (1,2) about the origin How did the coordinate change? 1 What matrix could you multiply 2 by to yield your new coordinate? Rotations about the origin of 270 o Rotate the point (1,2) about the origin How did the coordinate change? 1 What matrix could you multiply 2 by to yield your new coordinate? For a counterclockwise rotation about the origin of 90 o Rotation Matrices Symbolized by: Multiply the vertex matrix by: 180 o 270 o 12

13 Example 2 Use a reflection matrix to find the coordinates of a reflection over the y-axis of square SQAR with vertices S(4, 1), Q(7, 3), A(9, 0), R(6, -2). Then graph the pre-image and the image on the same coordinate grid. Example 3 An animated figure rotates about the origin. The image has key points at (5, 2), (3, -1), (2, -4), (-1, 2), and (2.5, 1.5). Find the locations of these points at the 90 o, 180 o, and 270 o counterclockwise rotations. Example 4 A parallelogram has vertices W(-2, 4), X(0, 8), Y(4, 6), and Z(2, 2). Find the coordinates of the dilated parallelogram W X Y Z for a scale factor of 1.5. Describe the dilation. 13

14 Section 5 Determinants and Multiplicative Inverses of Matrices Each square matrix has a. The determinant of number denoted by or det is a Second-Order Determinant Third-Order Determinant Example 1 Find the value of Example 2 Find the value of

15 Identity Matrix for Multiplication Identity Matrix for Second-Order Matrix Inverse Matrix A -1 Inverse of a Second-Order Matrix Example 3 Find the inverse of the matrix Example 4 Solve the system of equations by using matrix equations. 4x 2y = 16 x + 6y = 17 Example 5 A metallurgist wants to make 32 kilograms of an alloy with 60% iron. She has quantities of two metals, one with an iron content of 44% and another with an iron content of 92%. How much of each metal should she use? 15

16 Section 6 Solving Systems of Linear Inequalities Example 1 Belan Chu is a graphic artist who makes greeting cards. Her startup cost will be $1500 plus $0.40 per card. In order for her to remain competitive with large companies, she must sell her cards for no more than $1.70 each. How many cards must Ms. Chu sell in order to make a profit? Polygonal Convex Set Example 2 Solve the system of inequalities by graphing and name the coordinates of the polygonal convex set. x 0 y 0 x y 5 16

17 Vertex Theorem Example 3 Find the maximum and minimum values of f(x, y) = y 2x + 5 for the polygonal convex set determined by the system of inequalities. x 1 y 8 2x y 14 y 2 x y 5 Section 7 Linear Programming Linear 1. Define all variables. Programming Procedure 2. Write the constraints as a system of inequalities. 3. Graph the system and find the coordinates of the vertices of the polygon formed. 4. Write an expression whose value is to be minimized or maximized. 5. Substitute values from the coordinates of the vertices into the expression. 6. Select the greatest or least result. 17

18 Example 1 Suppose a lumber mill can turn out 600 units of product each week. To meet the needs of its regular customers, the mill must produce 150 units of lumber and 225 units of plywood. If the profit for each unit of lumber is $30 and the profit for each unit of plywood is $45, how many units of each type of wood product should the mill produce to maximize profit? Example 2 The profit on each set of cassettes that is manufactured by MusicMan, Inc., is $8. The profit on a single cassette is $2. Machines A and B are used to produce both types of cassettes. Each set takes nine minutes on Machine A and three minutes on Machine B. Each single takes one minute on Machine A and one minute on Machine B. If Machine A is run for 54 minutes and Machine B is run for 42 minutes, determine the combination of cassettes that can be manufactured during the time period that most effectively generates profit within the given constraints. 18

19 Example 3 The Woodell Carpentry Shop makes bookcases and cabinets. Each bookcase requires 15 hours of woodworking and 9 hours of finishing. The cabinets require 10 hours of woodworking and 4.5 hours of finishing. The profit is $60 on each bookcase and $40 on each cabinet. There are 70 hours available each week for woodworking and 36 hours available for finishing. How many of each item should be produced in order to maximize profit? Example 4 A manufacturer makes widgets and gadgets. At least 500 widgets and 700 gadgets are needed to meet minimum daily demands. The machinery can produce no more than 1200 widgets and 1400 gadgets per day. The combined number of widgets and gadgets that the packaging department can handle is 2300 per day. If the company sells both widgets and gadgets for $1.59 each, how many of each item should be produced in order to maximize profit? 19