systems of linear equations and inequalities. solving systems by graphing3.1
TRANSCRIPT
![Page 1: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/1.jpg)
Systems Of Linear Equations and Inequalities
![Page 2: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/2.jpg)
Solving Systems by Graphing—3.1
![Page 3: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/3.jpg)
System of two linear equations:
Solution: The ordered pair (x, y) that satisfies both equationsWhere the equations intersect
![Page 4: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/4.jpg)
Check to see if the following point is a solution of the linear system: (2, 2)
![Page 5: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/5.jpg)
Check to see if the following point is a solution of the linear system: (0, -1)
![Page 6: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/6.jpg)
Solving Graphically
![Page 7: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/7.jpg)
Solving Graphically
![Page 8: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/8.jpg)
Solving Graphically
![Page 9: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/9.jpg)
Solving Graphically
![Page 10: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/10.jpg)
InterpretationThe graphs intersect at 1 specific point
Exactly one solutionThe graph is a single line
Infinitely many solutionsThe graphs never intersect
No solutions
![Page 11: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/11.jpg)
p.142 #11-49 Odd
![Page 12: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/12.jpg)
Solving Systems Algebraically—3.2
![Page 13: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/13.jpg)
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
![Page 14: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/14.jpg)
Substitution Method
![Page 15: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/15.jpg)
Substitution Method
![Page 16: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/16.jpg)
Substitution Method
![Page 17: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/17.jpg)
Substitution Method
![Page 18: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/18.jpg)
p.152#11-19
![Page 19: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/19.jpg)
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
![Page 20: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/20.jpg)
Linear Combinations
![Page 21: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/21.jpg)
Linear Combinations
![Page 22: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/22.jpg)
Linear Combinations
![Page 23: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/23.jpg)
Linear Combinations
![Page 24: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/24.jpg)
p153 #23-31
![Page 25: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/25.jpg)
Pop Quiz!!Graphing Linear Inequalities
Graph the following:
![Page 26: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/26.jpg)
Graphing Linear Inequalities
![Page 27: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/27.jpg)
Graphing Linear Inequalities
![Page 28: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/28.jpg)
Graphing Linear Inequalities
![Page 29: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/29.jpg)
Solving Systems of Linear Inequalities—3.3
![Page 30: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/30.jpg)
SystemsSolution of two linear equations:
Ordered pair
Solution of two linear inequalitiesInfinite SolutionsAn entire region
![Page 31: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/31.jpg)
Solving Linear Inequalities
![Page 32: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/32.jpg)
Solving Linear Inequalities
![Page 33: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/33.jpg)
Solving Linear Inequalities
![Page 34: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/34.jpg)
Solving Linear Inequalities
![Page 35: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/35.jpg)
p.159 #13-49 EOO
![Page 36: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/36.jpg)
Optimization—3.4
![Page 37: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/37.jpg)
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
![Page 38: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/38.jpg)
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
![Page 39: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/39.jpg)
Solving: Finding min and maxObjective
Function:
Constraints:
![Page 40: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/40.jpg)
Objective
Function:
Constraints:
Solving: Finding min and max
![Page 41: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/41.jpg)
Objective Function:
Constraints:
Solving: Finding min and max
![Page 42: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/42.jpg)
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
![Page 43: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/43.jpg)
Writing InequalitiesOptimization:
Constraints:
![Page 44: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/44.jpg)
p.166 #9-15, 21
![Page 45: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/45.jpg)
Graphing in Three Dimensions—3.5
![Page 46: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/46.jpg)
z-axis
Ordered triple
Octants
![Page 47: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/47.jpg)
(-2, 1, 6)
![Page 48: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/48.jpg)
(3, 4, 0)
![Page 49: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/49.jpg)
(0, 4, -2)
![Page 50: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/50.jpg)
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
![Page 51: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/51.jpg)
Equations in 3 variables
![Page 52: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/52.jpg)
Equations in 3 variables
![Page 53: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/53.jpg)
Equations in 3 variables
![Page 54: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/54.jpg)
p.173 #22-33
![Page 55: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/55.jpg)
Solving Systems of Linear Equations in Three Variables—3.6
![Page 56: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/56.jpg)
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
![Page 57: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/57.jpg)
What does this look like graphically?
![Page 58: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/58.jpg)
![Page 59: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/59.jpg)
![Page 60: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/60.jpg)
![Page 61: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/61.jpg)
Should we solve graphicallyProbably not…
Tough to be accurateDifficult to find equations and coordinates in 3-
DSo….
Solve algebraically
![Page 62: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/62.jpg)
Solving Systems
![Page 63: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/63.jpg)
Solving Systems
![Page 64: Systems Of Linear Equations and Inequalities. Solving Systems by Graphing3.1](https://reader036.vdocument.in/reader036/viewer/2022062312/551a04a8550346634a8b4988/html5/thumbnails/64.jpg)
p.181 #12, 13, 17-20