2. The maximum or minimum value of ax + by cl
(where a, b, 0), on a region of the type
in Theorem 1 occurs at a vertex of the convex polygon enclosing that region..
育教僑華頁二第張六第日五十月三年區巴馬夏 WAH KIU YAT:PO.
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四期星日一月五年九六九一层公年八十五國民華中
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現代數學科 (廿六)
MODERN MATHEMATICS (26).
(Solutions of the exercises of last week
Solve:
4x-by=
5x + y = 23
Solution:
(5-3) (3) × (23)
-5
(4)(3)(3)79 (-3 (4)(23)
(3/3) - 금 (3)
19-19
Therefore, the set of all points of
{(x,y) y + 2x6 is the half-plane above the line defined by y + 2x - 6. The graph is the ahaded part as shown in the figure. Notice thos the line is not included in the graph
(2) Sketch the graph of y 21 46, universe being
Dr D.
Solution
Solution set- {(x,y) { y
Стара
+2x ≤6}
Employing the same route of reasoning, you wil bee that the graph of the solution set 18 the set of all points below the line defined by y + 2x 6y including the line itself. That is it is the other half-plane which is different from that of Example (1).
le shall accept the truthe or the theorems above without proofs.
Examples: (1) Compute an ordered pair of the region of
Example (3) in last section for wha on the value of the expression 5x + 6v 18. marymum Solution
Referring to Example (5) in section, we call the system of linear inequalities the system of constraints, and we call f(x,y) = 5x 6y the
bjective, function. We sketch again the graph or tha agatem in the accompanying figure.
2. Solve:
12x 53.13
1 * - 3y =
wolution:
2
1
(4:3)) - (3)
-1
193
-29 30-39
|
3. Solve
4
3y 22
2x - y 2z -1
X
Solution:
3
12
1
3월 3월 (1)
18
1
w
18
+
Z
18
1 ; Y = 1
2 = 1
+
4 Test whether each of the following is singular
or nonsingular.
(a)
(u)
2
(c) 2
小
?
2
Solution
(a) f
(b)
+
*
10
Bence all matrices
15
are nonsingular.
2
(3,0)
Vertices
c (2,0) D(-1,3) A. (-3,3) B (1-5)
f(x,y) = 5x
10 13.
бу
3 -45
You see that of all ordered pairs in the given region, the ordered pair (-1,3) yields the maximum value of the expression Sz + 6y. Observa that (-1.3) is one of the vertices of the nolygon ABCD,
(3) Sketch the graph of the system
x2-3
y 3
* y
D X D
x - y 42
Solution
Solution set
(2) Solve the system of linear inequalities. Then
find the ordered pair for which the linear expression is à maximum,....
Graphi
·
{(x,y) | = = =}}^{g} = }} n{(x,y)}x+y=2{0} {(x,y)} } x − y
421
The graph of the system is the spaded' region including its boundary.
T\X, J} - 20*
x64
x + 346
x + 572
(-3,3)
VIII Linear Programming
1. Linear inequalities and system of linear
inequalities:
In this section we are to Learn now to sketch the graphs of linear inequalities and system of
fhear inequalities in two variables. In the section that follows we shall introduos some concepts which will help us to solve samo physical problems by means of a method callou linear programming,
The graph of à gavep condition, equality of Finequality, is the set of all points whose
coordinates satisfy the given condition. The...
(4) Sketch the graph of the system
Jyo 1x56
Solutioni
12 - D x D
Solution . The graph, of the system of constrainto
Vértices
is shown in the accompanying figura.
Objective function (f(x,y)=2x+3y}
▲ (0.5)
15
B (0,2)
6
o (2,0)
4
D. (4,0)
8
E (4,2)
14
17
2+y=2
O
Bence at (1,5), ((x,y)
*
2x + 3y has the maximum
(3) Given záme me Example (2), find the ordered
pair for which 2x + 3y is a minimum. Solution↑
Using the table in Example (2), you see มอ ordered pair (2,0) yields the minimum value of ((x,y) - 2x + 3y.
X
3. Linear programming
In this section we bukas apply the preceduras of the previous sections to a kind of physical problems known as linear programming problems, which odour frequently in business and industry. The examples illustrate the method of solving simple linear programning problems.
(1) The number of unit. v. ZJ VONLUS A and B in one lb. of X and one lb. of Y are shown in the chart below, The minimum intake per day: i☐ 13 units of Vit. A and 14 units of Vit. B, What is the least total weight of these two substances which must be taken each day to
"
the graph of a condition is the graph of its solution set,
Examples:
(1) Sketch the graph of y 2x>6, universe being
"D x D.
Solution
Solution set - {(x, y)! y + `2x>6}
Notice that the miverse is the product set of real numbers. Hence the solution set of the condition cannot be tabulated, for between any two real numbers there are many numbers,
Graphi
We shall first draw the line defined by 7+ 21 - 6. Then the line divides the corrdinate plane into two half-planes, and you will see one of them must be the graph of the solution set. Let P be any point in the line. Through P we drav segment PQ parallel to y-axis. Obviously, tha first coordinates of P and Q are the same which .. but their second coordinates are different.
Solution set
=
Graph:
{(x,y) / x>o{n} {(x,y) { 7?°} < 8} n{(x,y) / x ≤6{n}(x, y) { x + y ≤ ← &
Since 0 and x6, the graph is between the linės defined by x = 0 and x 6. Since y≥0, the graph is above x-axis. The shaded region, including its boundary, is the graph required.
2. How to maximize and minimize the value a linear
expression, ax + by + e. under certain restrictions
In this section we are to learn how to maximize or minimize the value of a linear expression ax + by + c, in which the ordered pairs are members of a region whose boundary is a conver polygon. In more advanced mathematics there theorems which state the conditions under which the macımum or minimum value of the linear expression exists. The theoreme are stated as below.
are
1. If the graph of the solution set of a Synew of linear inequalities is a region of finita area, then the boundary, of the region is a.. convex polygon-
provide the intake?
Units of
Vit. A
Vit. B
3
5 4
Solution System or Constraints'
2x + 7y ⇒ 13
objective functions
5x 4y 14
10
y 20
Graphi
(未宪轉入第六張第三()