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Systems Of Linear Equations and Inequalities

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Solving Systems by Graphing3.1

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System of two linear equations: Solution: The ordered pair (x, y) that satisfies both equations Where the equations intersect

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Check to see if the following point is a solution of the linear system: (2, 2)

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Check to see if the following point is a solution of the linear system: (0, -1)

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Solving Graphically

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Interpretation The graphs intersect at 1 specific point Exactly one solution The graph is a single line Infinitely many solutions The graphs never intersect No solutions

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p.142 #11-49 Odd

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Solving Systems Algebraically3.2

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Substitution Method 1.) Solve one of the equations for one of the variables 2.) Substitute the expression into the other equation 3.) Find the value of the variable 4.) Use this value in either of the original equations to find the 2 nd variable

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Substitution Method

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p.152#11-19

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Solving by Linear Combination 1.) Multiply 1 or both equations by a constant to get similar coefficients 2.) Add or subtract the revised equations to get 1 equation with only 1 variable Something must cancel! 3.) Solve for the variable 4.) Use this value to solve for the 2 nd variable 5.) Smile

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Linear Combinations

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p153 #23-31

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Pop Quiz!! Graphing Linear Inequalities Graph the following:

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Graphing Linear Inequalities

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Solving Systems of Linear Inequalities3.3

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Systems Solution of two linear equations: Ordered pair Solution of two linear inequalities Infinite Solutions An entire region

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Solving Linear Inequalities

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p.159 #13-49 EOO

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Optimization3.4

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Optimization Finding the maximum or minimum value of some quantity Linear Programming: Optimizing linear functions Objective Function: What we are trying to maximize or minimize The linear inequalities making up the program: constraints Points contained in the graph: feasible region

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Optimal Solution The optimal Solution (minimum or maximum value) must occur at a vertex of the feasible region If the region is bounded, a minimum and maximum value must occur within the feasible region

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Solving: Finding min and max Objective Function: Constraints:

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Objective Function: Constraints: Solving: Finding min and max

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Objective Function: Constraints: Solving: Finding min and max

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A Furniture Manufacturer makes chairs and sofas from prepackaged parts. The table below gives the number of packages of wood parts, stuffing, and material required for each chair and sofa. The packages are delivered weekly and manufacturer has room to store 1300 packages of wood parts, 2000 packages of stuffing, and 800 packages of fabric. The manufacturer profits $200 per chair and $350 per sofa. How many of each should they make per week? MaterialChairSofa Wood4 boxes3 boxes Stuffing4 boxes3 boxes Fabric1 box2 boxes

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Writing Inequalities Optimization: Constraints:

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p.166 #9-15, 21

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Graphing in Three Dimensions3.5

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z-axis Ordered triple Octants

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(-2, 1, 6)

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(3, 4, 0)

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(0, 4, -2)

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Linear Equations ax + by + cz = d An ordered triple is a solution of the equation The graph of an equation of three variables is the graph of all its solutions -The graph will be a plane

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Equations in 3 variables

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p.173 #22-33

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Solving Systems of Linear Equations in Three Variables3.6

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Solutions 1 solution An ordered triple where all 3 planes intersect Infinite Solutions All 3 planes intersect to form a line No Solutions All 3 planes do not intersect All 3 planes do not intersect at a common point or line

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What does this look like graphically?

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Should we solve graphically Probably not… Tough to be accurate Difficult to find equations and coordinates in 3-D So…. Solve algebraically

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Solving Systems

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p.181 #12, 13, 17-20

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