1. 管理數學Chapter 2:System of Linear EquationsXX. XX, XXX
by
XXXX
2. AgendaLinear Systems as Mathematical Models
Linear Systems Having One or No Solutions
Linear Systems Having Many Solutions
3. Linear <-- Linea1 x + a2 y = b
or
a1 x1 + a2 x2 +....+ an xn = b
4. 2x + 3y = 4
x2 + y2 = 1
x1 - x2 + x3 - x4 = 6
z = 5 - 3x + y/2
sin x + ey = 1
xy = 2
7x1 + 3x2 + 9/x3 + 2x4 = 1
x1 + 2x2 + 3x3 +....+ nxn = 1Which one is linear?
5. Example 1A firm produces bargain and deluxe TV sets by buying the components, assembling them, and testing the sets before shipping.
6. ResourcesThe bargain set requires 3 hours to assemble and 1 hour to test. The deluxe set requires 4 hours to assemble and 2 hours to test. The firm has 390 hours for assembly and 170 hours for testing each week.
7. QuestionUse a system of linear equations to model the number of each type of TV set that the company can produce each week while using all of its available labor.
8. Problem FormulationDefine decision variables (unit of scale)
Define the linear relation between variables (write the linear equations)
9. Example 2A dietitian is to combine a total of 5 servings of cream of mushroom soup, tuna, and green beans, among other ingredients, in making a casserole.
10. Ingredient NutritionsEach serving of soup has 15 calories and 1 gram of protein, each serving of tuna has 160 calories and 12 gram of protein, and each serving of green beans has 20 calories and 1 gram of protein.
11. QuestionIf these three foods are to furnish 380 calories and 27 grams of protein in casserole, how many servings of each should be used?
12. Example 3A retailer has warehouses in Lima and Canton, from which two stores—one in Tiffin and one in Danville—place orders for bicycles. Tiffin orders 38 and Danville orders 46.
13. Limitations and QuestionEach warehouse has enough to supply all orders but twice as many are to be shipped from Lima to Danville as from Canton to Tiffin.
Write the linear equations.
14. AgendaLinear Systems as Mathematical Models
Linear Systems Having One or No Solutions
Linear Systems Having Many Solutions
15. Solving a System of Linear EquationsProblem formulation: variable definition and equations
Algorithms or formula
Interpretation of solutions
16. x + 3y = 9
-2x + y = -4{(1)-2x + y = 3
-4x + 2y = 2{(2)4x - 2y = 6
6x - 3y = 9{(3)System of 2 Linear Equations
18. A System of Linear Equations( A Linear System ) A finite collection of linear equations
a11 x1 + a12 x2 +....+ a1n xn = b1
a21 x1 + a22 x2 +....+ a2n xn = b2
....
am1 x1 + am2 x2 +....+ amn xn = bm
19. A SolutionTo a equation: a1x1 + a2x2 + .... + anxn = b
( t1, t2,...., tn )
To a linear system :
A solution to each of linear equation simultaneously
ps. Solution Set
20. Elementary Transformations1. Interchange the position of two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a multiple of one equation to another equation.
21. x1 + x2 + x3 = 2 --------(1)
2x1 + 3x2 + x3 = 3 ------(2)
x1 - x2 - 2x3 = -6 --------(3)Interchange (1) and (3).
Multiply (2) by 1/2.
Add a -1 multiple (2) to (1).Continuous Operations
34. Matrix of Coefficients1 1 1
2 3 1
1 -1 -2x1+x2+x3=2
2x1+3x2+x3=3
x1-x2-2x3=-6Coefficients of the system or Matrix A
35. Augmented Matrixx1+x2+x3=2
2x1+3x2+x3=3
x1-x2-2x3=-61 1 1 2
2 3 1 3
1 -1 -2 -6Coefficients and RHS, or [ A | B ].
36. Reduced Echelon Form1. Any rows with all zeros are at the bottom.
2. Leading 1.
3. Leading 1 to the right.
4. All other elements in a leading 1 column are zeros.
39. Elementary Row Operations1. Interchange two rows
2. Multiply the elements of a row by a nonzero constant
3. Add a multiple of the elements of one row to the corresponding elements of another row.
43. Equivalent SystemsSuppose that A and B are both systems of linear equations.
A and B are equivalent if they are related through elementary transformations.
A and B has the same solution if they are equivalent.
44. Solving a system of linear equationsGauss-Jordan Elimination
Gauss Elimination
45. Gauss-Jordan Elimination1. Write the augmented matrix.
2. Derive the reduced echelon form of the augmented matrix
3. Write the system of equations corresponding to the reduced echelon form.