7.5 Special Types of Linear Systems

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Presentation transcript:

7.5 Special Types of Linear Systems The equations you have been waiting for have finally arrived!

How many ways can you solve this? What strategies could you use? A farmer keeps track of his cows and chickens by counting legs and heads. If he counts 78 legs and 35 heads, how many cows and chickens does he have? How many ways can you solve this? What strategies could you use? What strategy will you use?

Let a = number of chickens Let c = number of cows Solve by using system of equations Let a = number of chickens Let c = number of cows a + c = 35; c = 35-a 2a + 4c = 78 2a + 4(35-a) = 78 2a + 140 – 4a = 78 -2a = -62 a = 31 c = 4

Special linear systems Answer the question: There are 4 cows and 31 chickens. A Consistent Independent System! Special linear systems Intersecting Parallel Same line One solution No solution Many solutions (x, y) 0 = 0 0 = 2 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 substitution or combination 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 Consistent and Dependent System Same line

Solve by substitution or combination then graph. False Statement Parallel lines Solve by substitution or combination then graph. 3x – 2y = 12 -6x + 4y = -12 Multiply top by 2 6x - 4y = 24 -6x + 4y = -12 0 = 12 (False)

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! Inconsistent System!

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