## Presentation on theme: "Advanced Algebra Chapter 3"— Presentation transcript:

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 equations Where 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

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

p.142 #11-49 Odd

Solving Systems Algebraically—3.2

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 2nd variable

Substitution Method

Substitution Method

Substitution Method

Substitution Method

p.152#11-19

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 2nd variable 5.) 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

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

Solving Linear Inequalities

Solving Linear Inequalities

Solving Linear Inequalities

Solving Linear Inequalities

p.159 #13-49 EOO

Optimization—3.4

Optimization 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

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

Solving: Finding min and max
Objective Function: Constraints:

Solving: Finding min and max
Objective Function: Constraints:

Solving: Finding min and max
Objective Function: Constraints:

A Furniture Manufacturer makes chairs and sofas from prepackaged parts
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 Fabric 1 box 2 boxes

Writing Inequalities Optimization: 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

Solutions 1 solution Infinite Solutions No Solutions
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

What does this look like graphically?

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

Solving Systems

Solving Systems

p.181 #12, 13, 17-20