6-1: Solving Systems by Graphing

Transcription

1 6-1: Solving Sstems b Graphing Objective: To solve sstems of linear equations b graphing Warm Up: Graph each equation using - and -intercepts sstem of linear equations: two or more linear equations solution: an ordered pair that makes all of the equations in a sstem true (the point of intersection) Eample 1: Is ( 1, ) a solution of the sstem of equations? Eample : Solve b graphing. Check our solution Eample : Solve b graphing. Check our solution

2 Eample 4: Solve b graphing. Check our solution Eample 5: Solve b graphing. Check our solution. 1 Eample 6: Solve b graphing. Check our solution Eample 7: Suppose ou have $400 in our bank account and are saving $50 each month. Your friend has $00 in her account and is saving $100 per month. Use a graph to predict when ou and our friend will have the same amount of mone in our accounts and how much mone ou will both have. Closure Question: 1 One equation in a sstem is 8. a) Write a second equation so that the sstem has one solution. b) Write a second equation so that the sstem has no solution. c) Write a second equation so that the sstem has infinitel man solutions.

3 6-: Solving Sstems Using Substitution Objective: To solve sstems of linear equations using substitution Warm Up: 1. Is (, 7) a solution to the sstem 1 and?. Solve the equation 5( ) for.. Solve the linear sstem b graphing. 1 Eample 1: Solve using substitution. Check our solution Eample : Solve using substitution. Check our solution Eample : Solve using substitution. Check our solution. 4 1

4 Eample 4: Solve using substitution. Check our solution Eample 5: Solve using substitution. Check our solution Eample 6: Solve using substitution. Check our solution If all variables cancel out: no solution false statement (parallel lines) infinite solutions true statement (same line) Eample 7: An office suppl compan sells two tpes of fa machines. The charge $150 for one of the machines and $5 for the other. The compan sold fa machines for a total of $900 last month. Write and solve a sstem of linear equations to determine how man of each tpe were sold. Closure Question: What is a good situation for using the substitution method? What is a difficult situation?

5 6-a: Solving Sstems Using Elimination Objective: To solve sstems of linear equations using elimination Warm Up: Solve each sstem b graphing Solve each sstem using substitution There is more than one wa to solve a sstem of equations. It is not alwas possible to determine an eact solution using the graphing method and it is not alwas reasonable to isolate one of the variables using the substitution method. In this case, the sstem can be solved using the elimination method. Eample 1: Solve using elimination. Check our solution Eample : Solve using elimination. Check our solution

6 Eample : Solve using elimination. Check our solution Eample 4: Solve using elimination. Check our solution Eample 5: Solve using elimination. Check our solution Eample 6: Suppose our communit center sells a total of 8 tickets for a basketball game. An adult ticket costs $.50. A student ticket costs $1. The sponsors collect $ in ticket sales. Write and solve a sstem of linear equations to determine the number of each tpe of tickets sold. Closure Question: Eplain when it is best to solve a sstem of equations using substitution and when it is best to solve using elimination.

7 6-b: Solving Sstems Using Elimination Objective: To solve sstems of linear equations using elimination Warm Up: Determine the opposite coefficient Simplif each epression. 4. ( 7 6) 5. (5 8) Solve the linear sstem using elimination Eample 1: Solve using elimination. Check our solution Eample : Solve using elimination. Check our solution. 5 5

8 Eample : Solve using elimination. Check our solution Eample 4: Solve using elimination. Check our solution Eample 5: Solve using elimination. Check our solution Eample 6: Shopping at Savers Mart, Lisa bus her children 4 shirts and pairs of pants for $ She returns the net da and bus shirts and 5 pairs of pants for $ Write and solve a sstem of linear equations to determine the price of each shirt and each pair of pants. Closure Question: When solving a linear sstem using elimination, how do ou determine the numbers used to multipl each equation?

9 6-4: Applications of Linear Sstems Objective: To write sstems of linear equations with real-world situations To choose the best method for solving a sstem of linear equations Warm Up: 1. Is ( 5, ) a solution of the sstem? Solve the sstem b graphing Solve the sstem using substitution Solve the sstem using elimination Was to Solve a Sstem of Linear Equations: Graphing (6-1): A useful method for approimating a solution, checking the reasonableness of a solution, and providing a visual model. Substitution (6-): A useful method when one of the variables has a coefficient of 1. Elimination (6-): A useful method when none of the variables has a coefficient of 1.

10 Eample 1: Suppose ou have a choice of two different internet service companies. The monthl charges for Compan A are $1 plus $.00 per hour. The monthl charges for Compan B are $7 plus $0.50 per hour. Write and solve a sstem of linear equations to determine the number of hours and the monthl charge for the services to be the same price. Eample : A compan sells brass and steel machine parts. One shipment contains 6 brass and 10 steel parts for the cost of $98. A second shipment contains 9 brass and 4 steel parts for the cost of $9. Write and solve a sstem of linear equations to determine the cost of each tpe of machine part. Eample : A group of 40 children attended a baseball game on a field trip. Each child received either a hot dog or a bag of popcorn. Hot dogs were $.5 and popcorn was $1.75. If the total bill was $8.50, write and solve a sstem of linear equations to determine how man hot dogs and bags of popcorn were bought. Closure Question: What things should ou consider when deciding which method to use to solve a linear sstem?

11 9-8: Sstems of Linear and Quadratic Equations Objective: To solve sstems of linear and quadratic equations Warm Up: Graph each quadratic equation Solve each quadratic equation b factoring You can solve sstems of linear and quadratic equations graphicall and algebraicall. This tpe of sstem can have two solutions, one solution, or no solutions. two solutions one solution no solution Eample 1: Solve each sstem b graphing. Check our solutions. a) 1 b) 4 5

12 Eample : Solve each sstem using substitution and factoring. Check our solutions. a) 8 b) c) 4 d) Closure Question: How man solutions can a sstem of linear and quadratic equations have? Draw an eample of each.

13 6-5: Linear Inequalities Objective: To graph linear inequalities in two variables To use linear inequalities when modeling real-world situations Warm Up: Check whether the ordered pairs are a solution of (, 7). (6, 9) Solve each inequalit for Eample 1: Check whether the ordered pairs are a solution of 1. (0, 0) (1, ) (, ) ( 4, 1) (4, 1) (, 1) ( 1, ) ( 1, ) Graph the inequalit and ordered pairs to show the relationship. Graphing a Linear Inequalit: 1) Graph the line Use a dashed line for < and > to show that the points on the line are not solutions. Use a solid line for and to show that the points on the line are solutions. ) Shade the correct half-plane Test a point to determine whether it is a solution of the inequalit. If the point is a solution, shade the half-plane it is in. If not, shade the other half-plane. Eample : Graph

14 Eample : Graph 4 1 Eample 4: Graph 5 Eample 5: Graph 4 9 Eample 6: Write a linear inequalit that represents the graph below. Eample 7: You budget $18 to spend on meat for a part. Ham costs $ per pound and turke costs $ per pound. Let represent the number of pounds of ham ou can bu. Let represent the number of pounds of turke ou can bu. Write an inequalit that shows the amounts of ham and turke ou can bu. Graph the inequalit. What are three possible amounts of each tpe of meat that can be bought within our budget? Closure Question: Eplain the difference between graphing linear inequalities and graphing linear equations.

15 6-6: Sstems of Linear Inequalities Objective: To solve sstems of linear inequalities b graphing To model real-world situations using sstems of linear inequalities Warm Up: Graph each linear inequalit solution of a sstem of linear inequalities: an ordered pair that makes all of the inequalities in the sstem true (the overlap in the shaded regions) Eample 1: Graph 5 1 Eample : Graph Eample : Graph 5 1

16 Eample 4: a) Graph 6 4 b) Describe the shape of the solution region. c) Determine the vertices of the solution region. d) Calculate the area of the solution region. Eample 5: Write a sstem of inequalities that represents the graph below. Eample 6: Suppose ou have two jobs, babsitting, which pas $8 per hour, and sacking groceries, which pas $10 per hour. You need to earn at least $10 per week, but can work no more than 18 hours. Draw a graph that shows how man hours ou can work at each job. Closure Question: How is the solution of a sstem of linear inequalities similar to the solution of a sstem of linear equations? How is it different?