Special Types of Linear Systems

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?

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.

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.

Graph the Equations Bravo Breakfast Crafty Catering 4x + 2y = 60 M U F I N S 50 40 30 20 10 Crafty Catering 0 5 10 15 20 25 30 2x + y = 30 2x + (0) = 30 2x = 30 x = 15 (15, 0) 2(0) + y = 30 y = 30 (0, 30) Cinnamon Rolls 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.

Special linear systems Intersecting Parallel Same Lines Lines Lines . One solution No solution Many solutions (x, y) 0 = 2 0 = 0 All variables cancel out and you end with a false statement All variables cancel out and you end with a true statement When you solve each system, you either get an ordered pair, a false statement, or both sides are equal.

Multiply the top equations by 2 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?????

Rewrite in slope-intercept form: y = mx + b 3x – 2y = 3 -6x + 4y = -6 y = 3/2x -3/2 You have the same equations, so you have the same line and infinite solutions! You can graph to check. Infinite solutions Same line

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

Solve by elimination then graph to check. 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?

Rewrite in slope-intercept form: 3x – 2y = 12 -6x + 4y = -12 y = 3/2x -6 y = 3/2x -3 Notice, same slope but different y-intercepts. You have parallel lines with NO solution. They will never intersect!

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

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?

Special linear systems: One More Time! Special linear systems: Intersecting Parallel Same line One solution No 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.