Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1.

Systems Of Linear Equations and Inequalities

Solving Systems by Graphing—3.1

System of two linear equations:

Solution: The ordered pair (x, y) that satisfies both equationsWhere the equations intersect

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

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

Solving Graphically

Solving Graphically

Solving Graphically

Solving Graphically

InterpretationThe graphs intersect at 1 specific point

Exactly one solutionThe graph is a single line

Infinitely many solutionsThe graphs never intersect

No solutions

p.142 #11-49 Odd

Solving Systems Algebraically—3.2

Substitution Method1.) Solve one of the equations for one of the

variables2.) Substitute the expression into the other

equation3.) Find the value of the variable4.) Use this value in either of the original

equations to find the 2nd variable

Substitution Method

Substitution Method

Substitution Method

Substitution Method

p.152#11-19

Solving by Linear Combination1.) Multiply 1 or both equations by a constant

to get similar coefficients2.) Add or subtract the revised equations to

get 1 equation with only 1 variable Something must cancel!

3.) Solve for the variable4.) Use this value to solve for the 2nd variable5.) Smile

Linear Combinations

Linear Combinations

Linear Combinations

Linear Combinations

p153 #23-31

Pop Quiz!!Graphing Linear Inequalities

Graph the following:

Graphing Linear Inequalities

Graphing Linear Inequalities

Graphing Linear Inequalities

Solving Systems of Linear Inequalities—3.3

SystemsSolution of two linear equations:

Ordered pair

Solution of two linear inequalitiesInfinite SolutionsAn entire region

Solving Linear Inequalities

Solving Linear Inequalities

Solving Linear Inequalities

Solving Linear Inequalities

p.159 #13-49 EOO

Optimization—3.4

OptimizationOptimization

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

Optimal SolutionThe 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

Solving: Finding min and maxObjective

Function:

Constraints:

Objective

Function:

Constraints:

Solving: Finding min and max

Objective Function:

Constraints:

Solving: Finding min and max

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?

Material Chair Sofa

Wood 4 boxes 3 boxes

Stuffing 4 boxes 3 boxes

Fabric 1 box 2 boxes

Writing InequalitiesOptimization:

Constraints:

p.166 #9-15, 21

Graphing in Three Dimensions—3.5

z-axis

Ordered triple

Octants

(-2, 1, 6)

(3, 4, 0)

(0, 4, -2)

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 it’s solutions

-The graph will be a plane

Equations in 3 variables

Equations in 3 variables

Equations in 3 variables

p.173 #22-33

Solving Systems of Linear Equations in Three Variables—3.6

Solutions1 solution

An ordered triple where all 3 planes intersectInfinite Solutions

All 3 planes intersect to form a lineNo Solutions

All 3 planes do not intersectAll 3 planes do not intersect at a common point

or line

What does this look like graphically?

Should we solve graphicallyProbably not…

Tough to be accurateDifficult to find equations and coordinates in 3-

DSo….

Solve algebraically

Solving Systems

Solving Systems

p.181 #12, 13, 17-20