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Special Types of Linear Systems

Special Types of Linear Systems. The Last Day of School!. For the last day of school the 8 th graders want to have breakfast catered. Two companies offer delicious breakfasts of bacon, eggs, hash browns, pancakes, and juice.

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Special Types of Linear Systems

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  1. Special Types of Linear Systems

  2. The Last Day of School! • For the last day of school the 8th graders want to have breakfast catered. • Two companies offer delicious breakfasts of bacon, eggs, hash browns, pancakes, and juice. • Bravo Breakfasts charges $50 to set up the meal and $10 per person. • Crafty Catering charges $75 to set up the meal and $10 per person. • Which company is the better deal?

  3. Write an Equation for each company and Graph it. • Bravo Breakfast: C = 50 + 10x • Crafty Catering: C = 75 + 10x • Which company is the better deal? • What do you notice about the two lines that is different from other graphs we have done? 300 200 100 C O S T 0 10 20 30 40 # of People The two lines are parallel. They will never intersect. Bravo Breakfast will always be the better deal.

  4. Last Day of School! • On the last day of school, students want to have the option of having cinnamon rolls (x) and muffins (y) with the breakfast. • Bravo Breakfast’s equation for this combination is: 4x + 2y = 60 • Crafty Catering’s equation for this combination is: 2x + y = 30 • Which company has the better deal? • Graph these equations using x- and y-intercepts.

  5. Graph the Equations Bravo Breakfast • 4x + 2y = 60 • 4x + 2(0) = 60 • 4x = 60 • x = 15 (15, 0) • 4(0) + 2y = 60 • 2y = 60 • y = 30 (0, 30) M U F F I N S 50 40 30 20 10 0 Crafty Catering 0 5 10 15 20 25 30 Cinnamon Rolls • 2x + y = 30 • 2x + (0) = 30 • 2x = 30 • x = 15 (15, 0) • 2(0) + y = 30 • y = 30 (0, 30) • Which company is the better deal? • What do you notice about the two lines? • The two lines are the same. Both companies offer the same deal on muffins and cinnamon rolls.

  6. Special linear systems IntersectingParallelSame line No solution One solution Many solutions (x, y) 0 = 2 0 = 0 When you solve each system, you either get an ordered pair, a false statement, or both sides are equal.

  7. Solve by elimination then graph to check. 3x – 2y = 3 -6x + 4y = -6 Multiply the top equations by 2 6x – 4y = 6 -6x + 4y = -6 0 = 0 (true) What does this mean?????

  8. Graph to check: 3x – 2y = 3 -6x + 4y = -6 What do we learn about these lines [0 = 0]? They are the same line.

  9. False Statement Parallel lines Solve by elimination then graph to check. 3x – 2y = 12 -6x + 4y = -12 Multiply top by 2 6x - 4y = 24 -6x + 4y = -12 0 = 12 (False) What does this mean?

  10. Graph to check: 3x – 2y = 12 -6x + 4y = -12 What do we learn about these lines [0 = 12]? They are parallel lines.

  11. Lines intersect at (2, 1) • Solve by elimination. • 2x + 3y = 7 -2x + 2y = -2 • 5y = 5 • y = 1 • Substitute and solve for x. • 2x + 3y = 7 • 2x + 3(1) = 7 • 2x = 4 • x = 2 What does this mean?

  12. One More Time! Special linear systems: IntersectingParallelSame line No solution One solution Many solutions (x, y) 0 = 2 0 = 0 When you solve each system, you either get an ordered pair, a false statement, or both sides are equal.

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