Chapter 6: Systems of Equations and Inequalities

Transcription

1 Chapter 6: Sstems of Equations and Inequalities 6-1: Solving Sstems b Graphing Objectives: Identif solutions of sstems of linear equation in two variables. Solve sstems of linear equation in two variables b graphing. Sstem of linear equations: Solution of a sstem of linear equations: Identifing Sstems of Solutions Tell whether the ordered pair is a solution of the given sstem. 1A: ( 5, ) 5 = 0 ; = 1 1a: ( 1, ) + = 5 ; + = 1 1B: (, ) + = 4 ; + = 1b: (, ) = 4-1 ; + = 6 All solutions of a linear equation are on its graph. To find a solution of a sstem of linear equations, ou need a point that each line has in common. In other words, ou need their point of intersection. Hint: Sometimes it is difficult to tell eactl where the lines cross when ou solve b graphing. It is good to confirm our answer b substituting it into both equations. Chapter 6 Page 1

2 Solving a Sstem Equations b Graphing Solve the sstem b graphing. Check our answer. A: = = a: = 1 = + 5 B: 1 = + = 4 Application Wren and Jenni are reading the same book. Wren is on page 14 and reads pages ever night. Jenni is on page 6 and reads pages ever night. After how man nights will the have read the same number of pages? How man pages will that be? Homework: Sec 6-1 (pg 86), 5, 9-1, 8, 9, -5 (5, 1, 1: Graph to solve and enter answer in online) Chapter 6 Page

3 6-: Solving Sstems b Substitution Objective: Solve sstems of linear equations in variables b substitution. Sometimes it is difficult to identif the eact solution to a sstem b graphing. In this case, ou can use a method called. The goal when using substitution is to reduce the sstem to that has onl. Then ou can solve this equation b the methods taught in Chapter. Step 1 Solving Sstems of Equations b Substitution Step Step Step 4 Step 5 Solving a Sstem of Linear Equations b Substitution Solve the sstem b substitution. 1A: = = 1a: = + = + 5 1B: = = 6 1b: = = 16 1C: + = 1 = 5 Chapter 6 Page

4 Sometimes ou substitute an epression for a variable that has a coefficient. When solving for the second variable in this situation, ou can use the Distributive Propert. Caution: When ou solve one equation for a variable, ou must substitute the value or epression into the other original equation, not the one that had just been solved. Using the Distributive Propert Solve b substitution : + 6 = 11 + = 5 a: + = 8 + = 9 Application One cable television provider has a $60 setup fee and $80 per month, and the second has a $160 equipment fee and $70 per month. a. In how man months will the cost be the same? What will that cost be? Homework: Sec 6-: (Pg 94) 1, 4, 8-16, 5, Chapter 6 Page 4

5 6-: Solving Sstems b Elimination Objectives: Solve sstems of linear equations in two variables b elimination. Compare and choose an appropriate method for solving sstems of linear equations. Another method for solving sstems of equations. Like substitution, the goal of elimination is to get that has onl. To do this b elimination, ou add the two equations in the sstem together. Remember that an equation stas balanced if ou add equal amounts to both sides. So, if 5 + = 1, ou can add 5 + to one side of an equation and 1 to the other side and the balance is maintained. Since and have opposite coefficients, the -term is eliminated. The result is one equation that has onl one variable: 6 = 18. When ou use the elimination method to solve a sstem of linear equations, align all in the equations. Then determine whether an like terms can be eliminated because the have. Step 1 Solving Sstems of Equations belimination Step Step Step 4 Eliminate Using Addition Solve b elimination. 1A: 4 = = 1a: + = = 14 Chapter 6 Page 5

6 When two equations each contain the same term, ou can subtract one equation from the other to solve the sstem. To subtract an equation add the opposite of each term. Elimination Using Subtraction Solve b elimination. A: + = 5 5 = 1 a: + = 15 + = 5 In some cases, ou will first need to multipl one or both of the equations b a number so that one variable has opposite coefficients. This will be the new Step 1. Elimination Using Multiplication First Solve the sstem b elimination. A: + = 11 + = 5 a: + = 6 + = B: 5 + = + = 10 b: + 5 = 6 4 = 5 Application Eample 4: Paige has $7.75 to bu 1 sheets of felt and card stock for her scrapbook. The felt costs $0.50 per sheet, and the card stock costs $0.75 per sheet. How man sheets of each can Paige bu? Write a sstem. Use f for the number of felt sheets and c for the number of card stock sheets. Homework: Sec 6-: (Pg 401) 1, 4, 7, 11-19, 47, 48 Chapter 6 Page 6

7 6-4: Solving Special Sstems Objectives: Solve special sstems of linear equations in two variables. Classif sstems of linear equations and determine the number of solutions. Consistent: Inconsistent sstem: Sstems with No Solution Solve. 1: = 4 + = 1a: = = 1 Method 1: Compares slopes and -intercepts Method 1: Method : Solve the sstem algebraicall. Use the substitution method because the first equation is solved for. Method : If two linear equations in a sstem have the same graph, the graphs are, or the same line. There are of the sstem because ever point on the line represents a solution of both equations. Chapter 6 Page 7

8 Sstems with Infinitel Man Solutions Solve A: = + + = 0 a: = = 0 Method 1: Compares slopes and -intercepts Method 1: Compares slopes and -intercepts Method : Solve the sstem algebraicall. Use the substitution method Method : Solve the sstem algebraicall. Use the elimination method. Consistent sstems can either be independent or dependent. Independent Sstem: Dependent Sstem: Chapter 6 Page 8

9 Classifing Sstems of Linear Equations Classif the sstem. Give the number of solutions. A: = + + = 1 1 a: + = 4 ( + ) = B: + = = b: = ( 1) = + C: = 4( + 1) = = 6 c: = Application Eample 4: Jared and David both started a savings account in Januar. If the pattern of savings in the table continues, when will the amount in Jared s account equal the amount in David s account? Use the table to write a sstem of linear equations. Let represent the savings total and represent the number of months. Homework: 6-4 (Pg 409) 1-9 odd, 1- Chapter 6 Page 9

10 6-5: Solving Linear Inequalities Objective: Graph and solve linear inequalities in two variables. Linear inequalit: Solution of a linear inequalit: Identifing Solutions of Inequalities Tell whether the ordered pair is a solution of the inequalit. 1A: (-, 4); < + 1 1a: (4, 5); < + 1 1B: (, 1); > - 4 1b: (1, 1); > - 7 A linear inequalit describes a region of a coordinate plane called a. All points in the region are solutions of the linear inequalit. The of the region is the graph of the related equation. Chapter 6 Page 10

11 Step 1 Step Step Graphing Linear Inequalities in Two Variables Graph the solutions of the linear inequalit. A: Step 1 The inequalit is alread solved for. Step Graph the boundar line =. Use a solid line for. Step The inequalit is, so shade below the line. Check: Graphing Linear Inequalities a: 4 > 1 Check b: 4 > 0 B: 5 + > -8 Step 1 Solved for. Step Graph. Use a dashed line. Step Shade Check Check: c: + 1 C: Check: Check Chapter 6 Page 11

12 Skip Eample Writing an Inequalit from a Graph Write an inequalit to represent the graph. 4A: 4a: 4B: 4b: Homework: 6-5 (Pg 418), 6, 11, 1-18, 0, 1, 4, 44, 54, 60 (Due in class periods on paper: 6, 15-18) Chapter 6 Page 1

13 6-6: Solving Sstems of Linear Inequalities Objective: Graph and solve sstems of linear inequalities in two variables. Sstem of linear inequalities: Solutions of a sstem of linear inequalities: Identifing Solutions of Sstems of Linear Inequalities Tell whether the ordered pair is a solution of the given sstem. 1A: ( 1, ) + 1 ; < + 1a: ( ) < + 0,1 ; 1 1B: ( 1, ) 5 ; < 1 + 1b: ( 0, 0) ; > + 1 > 1 To show all the solutions of a sstem of linear inequalities, graph the solutions of each inequalit. The solutions of the sstem are represented b the overlapping shaded regions. Below are graphs of Eamples 1A and 1B on p. 41. Chapter 6 Page 1

14 Solving a Sstem of Linear Inequalities b Graphing Graph the sstem of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. A: > + 5 a: + 1 > B: + < 4 + b: > In Lesson 6-4, ou saw that in sstems of linear equations, if the lines are parallel, there are no solutions. With sstems of linear inequalities, that is not alwas true. Chapter 6 Page 14

15 Chapter 6 Page 15 Graphing Sstems with Parallel Boundar Lines Graph the sstem of linear inequalities. A: + > 5 4 B: + < > 6 C: a: 1 + > b: c: > + > Homework: Sec 6-6 (Pg 44), 9, 16-18, -8, (Due in class periods on paper: 9, -8, )