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Unit 9: Systems of Equations

Unit 9: Systems of Equations. Miss Wheaton Math 8 Spring 2014. Warm Up. Solving Equations Review. What does it mean to solve an equation? Find the value(s) that will make the statement true What does the solution represent? All values that make the statement true.

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Unit 9: Systems of Equations

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  1. Unit 9: Systems of Equations Miss Wheaton Math 8 Spring 2014

  2. Warm Up

  3. Solving Equations Review • What does it mean to solve an equation? • Find the value(s) that will make the statement true • What does the solution represent? • All values that make the statement true

  4. Create equations that have the following solutions. x = 6 x = 0 x = -8 x = 15 x = 1/3 Work for 10 min Give your partner your equations only (not the solutions) Have them solve your equations to see if they get the correct solution. Compare and discuss results.

  5. Warm Up

  6. Graph your line and your partner’s line on the same x-y axis. Label and title your graph Be as precise and accurate as possible!

  7. Warm Up

  8. x + 6 = 20 1 solution Value that will make the statement true Infinite Solutions All values that will make the statement true

  9. Introduction to Systems system of equations – a set of two or more equations that contain two or more variables. Solution to a system of equations – set of values that are solutions of all the equations. (x,y) 3 ways to solve systems: • substitution • graphing • elimination

  10. Examples 1. y = x + 3 y = 2x + 5 2. y = 3x + 8 y = -7 + 2x

  11. OYO Solve the system and check your solution. 3. y = 4x - 6 y = x + 3 4. y = 2x + 8 y = -8 – 2x

  12. Closure Write down in your own words: 1. What is a system of equations, • What does the solution look like, • What does the solution represent?

  13. Warm Up 1. 3. 2. 4.

  14. Systems – Substitution 1. x – y = 3 x + 5y = 39 How is this different from yesterday? How is it the same? 2. 3x + y = 9 6x + 2y = 16

  15. OYO Solve the system and check your work! 3. x + 4y = -10 x – 3y = 11 4. –x + 10y = -8 x – 5y = 4

  16. Closure 2. 1.

  17. Warm Up 2)

  18. The Eagle Store has recently acquired three new items for purchase: DVD’s, movie tickets, and bags of popcorn to enjoy during the movies. At the store, they have bundled the items into three different packages as shown below. • Movie Lover’s Package: 2 DVD’s and 2 Tickets for $30 • Music Fanatic Package: 1 DVD and 2 Tickets for $25 • Movie Connoisseur’s Package: 2 Popcorns and 1 Ticket for $14 However, you are not interested in any of the packages but just the price of individual items. Find the price of each individual item.

  19. Closure 1.How are the strategies the groups used to solve similar? 2.How are the strategies the groups used to solve unique? 3.How could this problem be represented algebraically? 4.How can we apply the strategies discussed today to equations with variables?

  20. Warm Up Solve the equations. • 6x + 3 = x + 8 • 5a – 5 = 7 + 2a • 13x + 15 = 11x – 25 • 5t – 5 = 5t + 7 *How could you look at #4 and know that there is no solution? **How does solving these equations help you solve a system of equations?

  21. Functions Review Function – for each x there is 1 y. How can you tell if a table is a function?

  22. How can you tell if a graph is a function?

  23. Function Notation f(x) means “the function of x” f(x) = 3x + 2 We can also use other letters, such as g(x) or h(x).

  24. Examples: 1. If f(x) = 6x + 3 and g(x) = x + 8, solve f(x) = g(x). • Solve f(x) = g(x) when f(x) = 6x – 14 and g(x) = -8x OYO: • Solve f(x) = g(x) when f(x) = 3x – 1 and g(x) = 13 – 4x 4. Solve h(x) = d(x) when h(x) = 3(4x – 2) and d(x) = 12x

  25. Systems We know systems can have one solution. How do we write the solution to a system? Sometimes systems don’t have 1 solution. 5. Solve the system. 3x + 2y = 5 3x + 2y = 6 • How can you tell by looking at it that there is no solution?

  26. Systems - OYO Try to predict if the system will have 1 or no solution before you solve. 6. Solve the system. y = 3x + 8 y = -7 + 3x 7. Solve the system. y = 2x + 9 y = -8 + 2x

  27. Closing What does the solution look like when there is one solution to a system? No solutions?

  28. Systems by Substitution OFP Solve the system on your own sheet of paper. y = 3x + 8 y = x + 2

  29. Advanced Systems by Substitution OFP Solve the system on your own sheet of paper. 2x – 3y = -1 y = x - 1

  30. Warm Up Solve. • f(x) = g(x) when f(x) = 6x – 11 and g(x) = -3x + 5 • y = 6x – 11 -2x – 3y = -7 3. 2(3 + x) = 2x + 6

  31. GRAPH PAPER TOMORROW AND ALL WEEK!

  32. Examples 1. 3x + y = 8 6x + 2y = 16 OYO 2. -2x + 10y = -8 x – 5y = 4 3. y + 6x = 10 3y + 18x = 30

  33. Scavenger Hunt Guidelines: You will need 1 piece of paper, pencil, and calculator. You will solve that system (showing all work) and then go to the station that has that answer and solve the system there.

  34. Exit Ticket 1 of these systems has 1 solution, 1 has no solution, and 1 has infinite solutions. WITHOUT SOLVING the systems, specify which is which. 1. y = x + 2 2. x – y = 4 3. 2x + 4y = 10 y = 5x – 1 2x – 2y = 8 2x + 4y = 3

  35. Warm Up • Solve the equation. 9(3s + 7) = -34 – 6s • Solve the system. y = -3x + 5 5x – 4y = -3

  36. Substitution OFP Solve the systems. Show all work. • f(x) = g(x) when f(x) = 2(x + 3) and g(x) = 6 + 2x 2. 2x + y = 3 2x + y = 7

  37. Graphing Lines Review What is slope-intercept form? y = mx + b What is slope? m = y1 – y2 or m = rise x1 – x2 run What is y-intercept? b, where the graph crosses the y-axis

  38. Graphing Lines Review • y = 1/3 x – 2 • 3y = -9x + 3 OYO: 3. y = -2x + 6 4. ½ y = x – 2

  39. Warm Up Graph each equation on the same x-y axis. • y = x - 2 • y = -2x + 3 • Solve the system y = x+2 y = x – 3

  40. Systems by Graphing The solution to a system of equations is the ordered pair where the two graphs intersect.

  41. OYO

  42. Advanced • Where do the plans intersect? What does that mean in regards to your plan versus your partner’s? • Create a 3rd savings plan that can compete against the other two equations. Write a proposal to your parents or guardians to explain what they need to do to help you with the new plan and convince them that the new plan is what is best for you.

  43. Exit Ticket Graph the system by • substitution • graphing • Write down what you notice about the systems y = 2x – 1 y = x

  44. Warm Up • Solve the system by graphing and substitution y = 3x – 2 y = -1/2x + 1 2. What is 10% of 15? 3. You want to buy a pair of shoes that is normally $75 but is now 20% off. How much are the shoes? 4. How many minutes are in 2 hours?

  45. REGULAR Complete these problems that will help you with the task we are doing today. Show all work and write down all steps. • What is 15% of 40? • What is 75% of 20? • Find the cost of a shirt that is originally $15 and is 60% off. • Find the cost of headphones that are originally $55 but are on sale for 15% off. • Convert 7 hours into minutes. • Convert 0.4 hours into minutes

  46. Closure Discuss with your partner: What is the difference between solving a system by substitution versus graphing? Which do you prefer? Why?

  47. Warm Up

  48. At what point in solving these equations can you tell that they have no solution? • 3x + 4 = 3x – 5 • 4(f – 2) = 4f • -(4x – 10) = -5(x – 2) What would the solution to a system that had no solution look like on a graph?

  49. Graph this system of equations. y = ½ x + 1 y = ½ x – 3 What is the solution to this system? How can we tell this before we graph it? y = -3x + 4 y = -3x - 2

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