1. Systems of Linear Equations

2. Solving with graphs and substitution Use the following problems in this power point to answer these questions: Use the following problems in this power point to answer these questions: What does it mean to solve a system of linear equations? What does it mean to solve a system of linear equations? How can you use graphs to estimate the solution for a system of linear equations? How can you use graphs to estimate the solution for a system of linear equations? Explain how to use the substitution method to solve a system of linear equations. Explain how to use the substitution method to solve a system of linear equations. Once you have solved a system of equations, how can you check your solution? Once you have solved a system of equations, how can you check your solution?

3. Example 1a: Solve the following system of equations in two ways: by graphing and by substitution. Check your solution. Solve the following system of equations in two ways: by graphing and by substitution. Check your solution.

4. Example 1b (part 1) The Keasling Middle School club is planning a community event. Based on previous fund-raising events, they estimate that the event will be a sellout – filling all 300 seats in the school auditorium. Plans are to charge adults $8 and children $3 admission. The club wants to earn $2,000 from admission charges. The Keasling Middle School club is planning a community event. Based on previous fund-raising events, they estimate that the event will be a sellout – filling all 300 seats in the school auditorium. Plans are to charge adults $8 and children $3 admission. The club wants to earn $2,000 from admission charges.

5. Example 1b (part 2) a) Write an equation that expresses the relationship among adult attendance, child attendance, and the goal for income from admission charges. Explain what the variables represent. a) Write an equation that expresses the relationship among adult attendance, child attendance, and the goal for income from admission charges. Explain what the variables represent. b) Write an equation that expresses the relationship among number of adults, number of children, and total attendance. Explain what the variables represent. b) Write an equation that expresses the relationship among number of adults, number of children, and total attendance. Explain what the variables represent.

6. Example 1b (part 3) c) Show two different ways (graphing and substitution) you could find the number of adults and children at the event if the club is to meet their income goal of $2,000, and their attendance estimate of 300 people. c) Show two different ways (graphing and substitution) you could find the number of adults and children at the event if the club is to meet their income goal of $2,000, and their attendance estimate of 300 people. d) Solve the system of linear equations and check your solution. d) Solve the system of linear equations and check your solution.

7. Solving by elimination How can elimination of a variable be used to solve a system of linear equations? How can elimination of a variable be used to solve a system of linear equations? Example 2: What would be the steps in solving the system using the elimination method? Justify each step.

8. Example 3: (part a) A veterinarian needs 40 pounds of dog food that is 18% protein. He will combine a beef mix that is 25% protein with a chicken mix that is 5% protein. How many pounds of each does he need to make the 18% protein mix? A veterinarian needs 40 pounds of dog food that is 18% protein. He will combine a beef mix that is 25% protein with a chicken mix that is 5% protein. How many pounds of each does he need to make the 18% protein mix?

9. Example 3: (part b) a) Define your variables: explain what they represent. Write an equation that expresses the relationship among pounds of beef and pounds of chicken and the total pounds of the dog food. a) Define your variables: explain what they represent. Write an equation that expresses the relationship among pounds of beef and pounds of chicken and the total pounds of the dog food. b) Write an equation that expresses the relationship among the pounds of beef and pounds of chicken and the percentage of protein. b) Write an equation that expresses the relationship among the pounds of beef and pounds of chicken and the percentage of protein.

10. Example 3: (part c) c) Show two different methods of your choice to find the pounds of beef and the pounds of chicken needed if the veterinarian is to end up with a 40 pound bag of dog food that is 18% protein. c) Show two different methods of your choice to find the pounds of beef and the pounds of chicken needed if the veterinarian is to end up with a 40 pound bag of dog food that is 18% protein. d) Solve the system of linear equations using both methods, and explain the meaning of your ordered pair solution within this situation. Check your solution. d) Solve the system of linear equations using both methods, and explain the meaning of your ordered pair solution within this situation. Check your solution.

11. Which method to use? You have worked with three quite different methods for solving systems of linear equations. How do you decide on a method to use in any particular problem? Describe when one method may be better than another. You have worked with three quite different methods for solving systems of linear equations. How do you decide on a method to use in any particular problem? Describe when one method may be better than another. Is it possible to use any of the three methods when solving any system of two linear equations? Explain. Is it possible to use any of the three methods when solving any system of two linear equations? Explain.

12. Systems with zero and infinitely many solutions What does it mean to say that a system of linear equations has no solutions? Infinitely many solutions? What does it mean to say that a system of linear equations has no solutions? Infinitely many solutions? How can you tell from a graph that a system of linear equations has no solutions? Infinitely many solutions? How can you tell from a graph that a system of linear equations has no solutions? Infinitely many solutions? How can you decide whether a system of linear equations has no solutions, exactly one solution, or infinitely many solutions just by examining the equations? How can you decide whether a system of linear equations has no solutions, exactly one solution, or infinitely many solutions just by examining the equations?

13. If a system of linear equations has infinitely many solutions, does that mean that any ordered pair (x, y) is a solution? Why or why not? If a system of linear equations has infinitely many solutions, does that mean that any ordered pair (x, y) is a solution? Why or why not?

14. Example 4a: Find values of a and b so that the system has infinitely many solutions. Find values of a and b so that the system has infinitely many solutions. Explain how the graphs of the equations will be related. Explain how the graphs of the equations will be related.

15. Example 4b: Find all values of a and b that will make the system have no solutions. Find all values of a and b that will make the system have no solutions. Explain how the graphs of the equations will be related. Explain how the graphs of the equations will be related.

16. Example 4c: Find all values of a and b that will make the system have exactly one solution. Find all values of a and b that will make the system have exactly one solution. Explain how the graphs of the equations will be related. Explain how the graphs of the equations will be related.

17. Assignment Using this power point as a guide, write a paper on the concepts and methods developed in this power point. Answer all questions and problems. Justify answers and explain your work as if you were teaching someone a lesson on systems of equations who has not seen this power point. Using this power point as a guide, write a paper on the concepts and methods developed in this power point. Answer all questions and problems. Justify answers and explain your work as if you were teaching someone a lesson on systems of equations who has not seen this power point. The paper should be typed and graphs made by an electronic graphing utility. The paper should be about 5 to 7 pages long, and double spaced. The paper will be assessed against the following assessment criteria: The paper should be typed and graphs made by an electronic graphing utility. The paper should be about 5 to 7 pages long, and double spaced. The paper will be assessed against the following assessment criteria: A) Knowledge and Understanding and C) Communication