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Advanced Algebra Chapter 3. 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.

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advanced algebra chapter 3

Advanced Algebra Chapter 3

Systems Of Linear Equations and Inequalities

slide3

System of two linear equations:

  • Solution: The ordered pair (x, y) that satisfies both equations
    • Where the equations intersect
interpretation
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
substitution method
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
solving by linear combination
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 
systems
Systems
  • Solution of two linear equations:
    • Ordered pair
  • Solution of two linear inequalities
    • Infinite Solutions
    • An entire region
optimization
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
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
Solving: Finding min and max
  • Objective Function:
  • Constraints:
slide40

Solving: Finding min and max

  • Objective Function:
  • Constraints:
solving finding min and max41
Solving: Finding min and max
  • Objective Function:
  • Constraints:
slide42

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?

writing inequalities
Writing Inequalities
  • Optimization:
  • Constraints:
slide46

z-axis

  • Ordered triple
  • Octants
linear equations
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

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