Chapter 4: Systems of Equations and Inequalities


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1 Chapter 4: Systems of Equations and Inequalities
2 4.1 Systems of Equations A system of two linear equations in two variables x and y consist of two equations of the following form: Equation 1: ax + by = c Equation 2: dx + ey = f where the solution (x,y) satisfies both equations. Checking Solutions of a Linear System: 3x 2y = 2 x + 2y = 6 1.) Is (2,2) a solution of the above system of equations? 2.) Is (0,1) a solution of the above system of equations? Solving a System Graphically :
3 Examples: 1.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator. 2x 2y = 8 2x + 2y = 4 Check Algebraically.. 2.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator. 3x 2y = 6 3x 2y = 2 3.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator. 2x 2y = 82x + 2y = 8
4 Solving a System by Substitution Solve each system below by the method of Substitution. y = 3x 3 1) y = x + 5 2) x + y = 3 3x + y = 1 Solve the system below by the method of Substitution, demonstrating that there is no solution. 3) 2x 2y = 0 x y = 1 What does the graph of this system look like? Solve the system below by the method of Substitution, demonstrating that there are infinitely many solutions. 4) x + y = 7 2x = 14 2y What does the graph of this system look like?
5 Applications Use your graphing calculator to graph the system of equations for each application below and to answer related questions. Create a sketch of each graph, labeling the axes with appropriate scales. 1) You are checking out cell phone plans and discover that Talk Anytime Wireless charges $50.00 per month for the first phone line and charges $20.00 per additional phone line. Text Away Wireless charges $80.00 per month for the first phone line and $5.00 per additional phone line. Determine the number of additional phone lines for which it would be cheaper to use Talk Anytime verses Text Away. 2) James and Zach began saving money from their parttime jobs. James started with $50 in his savings and earns $10 per hour at his job. Zach started with $225 in his savings and earns $7.50 per hour. If both boys save all of their earnings (and we disregard tax) when will they have the same amount of savings? 3) You are choosing between two movie rental services. Company A charges $2.99 per movie plus a $20 monthly fee. Company B charges $4.99 per movie with no monthly fee. How many movies could you rent and get charged the same monthly bill? If you only rent, on average, 8 movies per month, which is the better deal for you?
6 Check for Understanding 1) You are checking a solution of a system of linear equations. How can you tell if the solution is valid or not? 2) Describe how the graph of a system of linear equations looks when a. There is not solution. b. There is exactly one solution. c. There are infinitely many solutions.
7 4.2 Linear Systems in Two Variables Solving a system by the method of Elimination 1. 3x + 2y = 4 5x 2y = 8 4x 5y = x y = x + 9y = 8 2x + 6y = 7
8 Applications 1. A bus station 15 miles from the airport runs a shuttle service to and from the airport. The 9:00 a.m. bus leaves for the airport traveling 30 mph. The 9:05 a.m. bus leaves for the airport traveling 40 mph. Write a system of linear equations to represent distance as a function of time for each bus. How far from the airport will the 9:05 a.m. bus catch up to the 9:00 a.m. bus? D = 30t D = 40 t The school yearbook staff is purchasing a digital camera. Recently the staff received two ads in the mail. The ad for store #1 states that all digital cameras are 15% off. The ad for store #2 gives a $300 coupon to use when purchasing any digital camera. Assume that the lowest priced digital camera is $700. When could you get the same deal at either store? Let C = the cost of a camera after the discount Let x = the original cost of a camera
9 3. You are starting a business selling boxes of handpainted greeting cards. To get started, you spend $36 on paint and paintbrushes that you need. You buy boxes of plain cards for $3.50 per box, paint the cards, and then sell them for $5 per box. How many boxes must you sell for your earnings to equal your expenses? What will your earnings and expenses equal when you break even? (Write an equation to represent Total Expenses and another equation to represent Total Earnings.) 4. You commute to center city 5 days per week on a SEPTA train. You can purchase a monthly pass for $140 per month or purchase a round trip ticket each day that you commute for $9.50 per ticket. What is the number of days that you must ride to begin saving money by using the monthly pass? C = the cost in $ x = the number of days commuting
10 5. A soccer league offers two options for membership plans. Option A: an initial fee of $40 and then you pay $5 for each game that you play. Option B: you have no initial fee but must pay $10 for each game that you play. After how many games will the total cost of the two options be the same?
11 4.3 Linear Systems in Three Variables In addition to systems of two equations, it is sometimes necessary to solve a system of 3, 4 or more equations in 3, 4 or more variables. In this lesson we will learn to solve such systems algebraically. Later in the chapter we will use a matrix and our graphing calculator to solve such systems. Back Substitution This example has a reasonably straightforward set up allowing us to use simple back substitution to solve. x 2y + 2z = 9 y + 2z = 5 z = 3 Method of Elimination This example requires that we eliminate x by combining Equations 1 and 2, and also eliminate x by combining Equations 2 and 3. We can now use the Elimination method to solve the resulting equations for y and z, and then back substitute to solve for x. Example 1: x 2y + 2z = 9 Equation 1 x + 3y = 4 Equation 2 2x 5y + z = 10 Equation 3
12 Depending on the set up of the system, you may wish to eliminate y or z from the original pairs of equations. Example 2: 4x + y 3z = 11 2x 3y + 2z = 9 x + y + z = 3 How many solutions are possible?? The graph of a system of 3 linear equations in 3 variables consists of 3 planes. The planes may intersect in one point, in one line, in one plane or not at all.
13 An Inconsistent System: x 3y + z = 1 2x y 2z = 2 x + 2y 3z = 1 A System with Infinitely Many Solutions: x + y 3z = 1 y z = 0 x + 2y = 1
14 Let s look at these applications from your textbook.
15 4.4 Matrices and Linear Systems and 4.5 Determinants and Linear Systems (Day 1) Matrix Operations Always read a matrix ROW by COLUMN # Rows: Dimension: # Columns: Numbers in the matrix are called entries. What is the entry in the 2 nd row and 3 rd column for the matrix above? Different Types of Matrices Name Example Dimensions Row Matrix x 4 Column Matrix x 1 Square Matrix x 3 What are the dimensions of each matrix below?
16 Matrix Addition and Subtraction: Matrices must have the SAME dimensions. Add or subtract the corresponding entries. Example 1: Dimension of each matrix: Dimension of the answer matrix: Example 2: = Dimension of each matrix: Dimension of the answer matrix: Scalar Multiplication: Multiply the constant OUTSIDE the matrix to EACH entry inside the matrix. Example 3: = Dimension of the answer matrix: Scalar Multiplication combined with Addition or Subtraction: Example 4: = Dimension of each matrix: Dimension of the answer matrix: Solve the following matrix for x and y Corresponding entries are equal Example 5: 2 3x y =
17 4.4, 4.5 Homework Day 1: Matrix operations Perform the indicated operation if possible. If not possible, state the reason = = = = Solve the matrix equation for x and y x 10 = y y = 3 7 x y x 2 = y x 3 =
18 (Day 2) Matrix Multiplication: The number of columns in the first matrix must match the number of rows in the second matrix. If [A] has dimensions m x n If [B] has dimensions n x p The product of [A]x[B] will have dimensions m x p A: 2 X 3 B: 3 X 4 A: 3 X 2 B: 3 X 4 Dimension of [A]x[B]: Dimension of [A]x[B]: Example 6: Find AB A = B = Dim of [A]: Dim of [B]: Product Dim: Example 7: Find BA A = B = Dim of [A]: Dim of [B]: Product Dim: Example 8: Find AB + BC A = , B = , and C =
19 Use your calculator to add, subtract, multiply with matrices. To enter a Matrix in your calculator: 2 nd MATRIX EDIT ENTER (enter the dimensions of the matrix and the entries) To call up a Matrix in your calculator from the home screen: 2 nd MATRIX (highlight the matrix) ENTER A = , B = , and C = ) B (A + C) 2.) BA + BC Application of Matrices: A health club offers three different membership plans. With Plan X, you can use all club facilities: the pool, fitness center, and racket club. With Plan Y, you can use the pool and fitness center. With Plan Z, you can only use the racket club facilities. The matrices below show the annual cost for a Single and a Family membership for the years 2012 through [A] [B] [C] single family single family single family plan X plan Y plan Z plan X plan Y plan Z plan X plan Y plan Z ) Determine a matrix that gives the price increase from 2012 to 2014 for each of the plans. 2) Determine a matrix that gives the total cost for all three years for each of the plans. 3) The health club offered a 3year membership based on the 2012 rates. How much money does the 3year membership save for each plan compared to paying the regular membership rate for each of the 3 years?
20 Homework 4.4, 4.5 Day 2: Multiplying matrices For the matrices with the given dimensions, what are the dimensions of the product? If the product is undefined, explain why. 1. A: 2 X 5 B: 5 X 3 2. A: 6 X 2 B: 3 X 1 Dimension of [A]x[B]: Dimension of [A]x[B]: 3. A: 3 X 1 B: 1 X 2 4. A: 1 X 6 B: 6 X 1 Dimension of [A]x[B]: Dimension of [A]x[B]: Write the product. If it is not defined, state the reason = = = = Given matrices A, B and C, determine the products. If the product is not defined, state the reason B = 5 A = C = [A][B] = 10. [A][C] = 11. [C][B] = 12. [B][C] = 13. [C][A] = 14. [B][A] =
21 (Day 3) Use a matrix and a graphing calculator to solve a linear system 2 nd MATRIX EDIT ENTER (edit matrix) 2 nd MATRIX MATH B rref( 2 nd MATRIX (select the matrix that you edited) 1) 2x y + 4z = 48 x + 2y + 2z = 6 x 3y + 4z = 54 Use the matrix: Solution matrix: x y z 2) x + y 2z = 9 2x + y + z = 0 x 2y + 6z = 21
22 Homework 4.4, 4.5 Day 3: Use a matrix to solve a system Solve the system of equations using a matrix. 1. 9x + 8y = 6 x y = 1 2. x 3y = 2 5x + 3y = x y 4z = 3 x + 3y z = 1 x y + 5z = x + 10y z = 3 11x + 28y 4z = 16x 15y + 2z = x 3y + 5z = 1 3x + 2y + 4z = 11 2x y + 3z = 4
23 (Day 4) Determinants Determinant of a 2 x 2 matrix: a b a b det = c d = ad bc c d Determinant of a 3 x 3 matrix: a b c det d e f = (aei + bfg + cdh) ( ceg + afh + bdi) g h i Evaluate a determinant in your calculator: 1) Enter the determinant as a matrix: 2 nd MATRIX EDIT ENTER 2 nd QUIT (enter the dimensions of the matrix and the entries) 2) Evaluate the determinant: 2 nd MATRIX MATH 1:det( 2 nd MATRIX Select the matrix that you edited. ENTER Check the value of the determinants above by using your calculator.
24 Homework 4.4, 4.5 Day 4: Determinants and systems Evaluate the determinant = = Use Matrices to solve the system of equations x + y z = 3 2x 3y + 4z = 23 3x + y 2z = 15 3x + 3y + 4z = 1 3x + 5y + 9z = 2 5x + 9y +17z = 4 5x + 3y 2z = 4 2x + 2y + 2z = 0 3x + 2y +1z = 1 2x 4y + 5z = 33 4x y = 5 2x + 2y 3z = 19 Applications: 1. Claire and Dale shopped at the same store. Claire bought 5 kg of apples and 2 kg of bananas and paid altogether $22. Dale bought 4 kg of apples and 6 kg of bananas and paid altogether $33. Use matrices to find the cost of 1 kg of bananas.
25 2. Ann and Billy both entered a quiz. The quiz had twenty questions and points were allocated as follows: P points were added for each correctly answered question Q points were deducted for each incorrect (or unanswered) question Ann got 15 questions correct and scored 65 points. Billy got 11 questions correct and scored 37 points. Use matrices to find the value of Q. 3. A community relief fund receives a large donation of $2800. The foundation agrees to spend the money on $20 school bags, $25 sweaters, and $5 notebooks. They want to buy 200 items and send them to schools in earthquakehit areas. They must order as many notebooks as school bags and sweaters combined. How many of each item should they order? 4. An ultimate Frisbee team has to order jerseys, shorts, and hats. They have a budget of $1350 to spend on $50 jerseys, $20 shorts, and $15 hats. They want to buy 40 items in preparation for the oncoming season and must order as many jerseys as shorts and hats combined. How many of each item should they order?
26 4.6 Systems of Linear Inequalities Graph the following systems of inequalities and label the vertex/vertices: 10 1) y 3x 1 y < x ) x 0 y 0 x y
27 3.) x < y x + 3y < 9 x ) x + 2y 10 2x + y 8 2x 5y <
28 Write the system of inequalities that correspond with the shaded region.
29 4.6 HOMEWORK Graph the system of linear inequalities. 1) y > 2 y 1 2) y > 5x x 5y ) x y > 7 2x + y < 8 4) y < 4 x > 3 y > x
30 5) 2x 3y > 6 5x 3y < 3 x + 3y > 3 6) y < 5 y > 6 2x + y 1 y x Challenge. Write a system of linear inequalities for the region.
31 Review Worksheet for Chapter 4 Test Complete the following problems from the ebook: p (9, 11, 15, 17, 23, 27, 31, 33, 37, 39, 41, 43, 71) Complete the following problems with matrices.
32
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