Why do we need Systems of Equations?

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Presentation transcript:

Why do we need Systems of Equations?

Why do we need this weird math?? Systems of equations can be amazingly useful. How else would you figure out how many ounces of 70% dark chocolate and 20% milk chocolate you need to mix to get one pound (16 oz) of 40% chocolate? We forgot to mention: the meaning of life involves mixing chocolate. There. Now you know. While problems with two unknowns may seem impossible to solve at first, we can use a system of equations to organize our information, then use one of three methods to solve the system and find the answer. Oh, organization. You're a cure for all ills.

Unit 6a Will focus on solving systems of equations by …. GRAPHING & Finding the Intersection Point!

Notes: There are 3 Types of solutions Infinite Solutions: If a system has at least 1 solution, it is said to be CONSISTENT. The graphs intersect at one point or are the same line Exactly One Solution: If a consistent system has exactly one solution, it is said to be INDEPENDENT. If it has an infinite number of solutions, it is DEPENDENT. No Solution: If a system has no solution, it is said to be inconsistent. The graphs are parallel.

NOTES: Example 1 Use a graph to determine whether the system is consistent or inconsistent and if it is independent or dependent. y = –x + 1 y = –x + 4 Answer: The graphs are parallel, so there is no solution. The system is inconsistent.

Example 2 Use a graph to determine whether the system is consistent or inconsistent and if it is independent or dependent. y = x – 3 y = –x + 1 Answer: The graphs intersect at one point, so there is exactly one solution. The system is consistent and independent.

Quick Check Use the graph to determine whether the system is consistent or inconsistent and if it is independent or dependent. 2y + 3x = 6 y = x – 1 A. consistent and independent B. inconsistent C. consistent and dependent D. cannot be determined

Quick Check Use the graph to determine whether the system is consistent or inconsistent and if it is independent or dependent. y = x + 4 y = x – 1 A. consistent and independent B. inconsistent C. consistent and dependent D. cannot be determined

NOTES: Example 3 Graph each system and determine the number of solutions that it has. If the system has one solution, name it. y = 2x + 3 8x – 4y = –12 Answer: The graphs coincide. There are infinitely many solutions of this system of equations.

LAST QUICK CHECK! Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. A. one; (0, 3) B. no solution C. infinitely many D. one; (3, 3)