Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Mathematics in Modern World
(1RDT-A) | BATCH 2025
TRANSCRIBED BY: Dumanon, Enriquez, Maraggun
LECTURER/S: Mr
...

REFERENCES: PPT, Handouts, Internet, textbook

↣ To graph a linear

Topic Outline
1 Linear Programming

inequality in two variables

2 Systems of Linear Inequalities

(say, x and y ),
↣ first get y alone on one

3 Modeling with Linear Programming
4 Graphical Method

side
...


A modeling procedure that is used to

↪ If the inequality is not

maximize profits, minimize costs, control
inventories, efficiently assign tasks to

strict ( ≤ and ≥ ), graph

personnel, etc
...

Difference 𝑦 = 4𝑥 + 2

Linear
Equation
Linear
Inequality

↣ An equation that has the
form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
...
)

inequality

𝑦 ≤ 4𝑥 + 2
(kapag sinabi naman na
inequality ang makukuha mo
namang final answer is pwedeng
less than, greater than, less than
or equal to or greater than or
equal to sa given
...

(kapag may underline yung less than or
greater than, solid line yung dapat na
makikita sa graph)

Example 4:
Graph the inequality 𝑥 > 2
...

(ibig sabihin nung 𝑥 = 2
...
5 is yung
Example 2:

shaded part naman yung tinutukoy)

Graph the inequality 𝑥 + 𝑦 > −5
...

Example 3:

↪ {

Graph the inequality 𝑦 ≤ 2
...

↪ The area in which all the inequalities
overlap
...


∴ Note: If there is no area in which all the
inequalities overlap, then the system has
no solution
...


{

𝑦 < 2𝑥 − 3
1

𝑦 ≤ 2𝑥 + 1

𝑥 + 4𝑦 ≥ 4
{ 𝑥+𝑦 ≤4
2𝑥 − 𝑦 ≥ −1

The solution is a region containing an
infinite number of ordered pairs
...


So:

2 Write the system of inequalities
...


↪ 8𝑥 + 4𝑦 = −4

4 Find the coordinates of the vertices of

(After maidistribute yung 4, proceed naman
sa pagcocombine like terms)
↪ 9x + 0 = 0
↪ 𝑥=0
(kaya zero kasi kapag lahat g dinidivide sa
zero is zero parin yung sagot)
(after hanapin si 𝑥 gagamitin yung value nya

the feasible region
...

6 Substitute the coordinates of the vertices
into the function
...


para sa 𝑦)
↪ 2𝑥 − 𝑦 = −1

Example 8:

↪ 𝑦 = 2𝑥 + 1(transpose yung 2𝑥 to the

A small furniture shop makes tables and

other side, then isubstitute na yung

chairs which must be processed through

assembly and finishing processes
...
e
...

↪ Making one table requires 4 hours to
assemble and 2 hours to finish
...

(Gamit tayo ng GeoGebra para malaman
yung graph, so feasible region which is
in quadrant 1 where x ≥ 0 and y ≥ 0
...

↪ One table contributes ₱ 500 to the
profit, while a chair contributes ₱ 300
...


sold
...

Let 𝑥 = number of tables = 4 hours and 2
hours
Let 𝑦 = number of chairs = 2 hours and
4 hours
2 Write the system of inequalities
...
)
5 Write the function to be maximized or
minimized
...

(𝑥, 𝑦)

500𝑥 + 300𝑦

𝑓(𝑥, 𝑦)

(0,0)

500(0) + 300(0)

0

(0,12)

500(0) + 300(12)

3,600

(12,6)

500(12) + 300(6)

7,800

𝑧𝑒𝑟𝑜
...


(15,0)

500(15) + 300(0)

7,500

𝑍 − 500𝑥 − 300𝑦 = 0

7 Select the greatest or least result, based

3

Construct a simplex tableau
...


exception of the last column, the

The maximum profit (₱ 7,800) is

numerical values given below are the

achieved when 12 tables and 6 chairs are

coefficients of each variable
...


example, in the first equation, the
coefficient of 𝑥 is 4, the coefficient of 𝑦

Step by step solution for example 8:

is 2, and the coefficient of 𝑆1 is 1
...


↪ recall: this is the same as the profit
function 𝑓(𝑥, 𝑦) = 500𝑥 + 300𝑦
Subject to:
↪ 4𝑥 + 2𝑦 ≤ 60 (Explicit constraint)

4

Find the column that contains the most
negative number
...


↪ 𝑥, 𝑦 ≥ 0 (implicit constraints)
(Note: Make sure that the inequality sign of
the explicit constraints is less than or equal
to
...
Use the values

to, while minimization involves greater than

in column 1 (the location of the most

or equal to
...
Find the ratio of the given

constraints) into equations and add a

values
...

↪ 4𝑥 + 2𝑦 + 𝑆1 = 60

↪

60
4
48
2

= 15
= 24

↪ 2𝑥 + 4𝑦 + 𝑆2 = 48
2

Rearrange the objective function, i
...
,
profit function, such that it is equal to

Compare the calculated values and
choose the lowest value
...


6

Transform the pivot element into 1
...
Find the ratio of the given

do that, multiply the first row (𝑅1 ) by 4,

values
...
e
...


↪
↪

15
1⁄2
18
3

= 30

=6

Compare the calculated values and
7

Refer to the column that contains the

choose the lowest value
...
Transform all the

the pivot element is equal to 3
...

↪ For the second row (𝑅2 ), perform
the operation 𝑅2 − 2𝑅1
...
To
do that, multiply the second row (𝑅2 )
1

by 3, i
...
, the reciprocal of the pivot
element
...

11 Refer to the column that contains the
pivot element
...
So, we

column into 0
...
(We only stop if
there are no more negative values in the
bottom row
...
It is actually column

↪ For the first row (𝑅1 ), perform the
1

operation 𝑅1 − 2 𝑅2
...


2
...
Use the values
in column 2 (the location of the most
negative number) and the values in the

negative numbers in the bottom row
...

↪ Number of tables: 𝑥 = 12

↪ Number of chairs: 𝑦 = 6

𝑃𝑖𝑣𝑜𝑡 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 = 4

↪ Maximum Profit: 𝑍 = 7, 800

X

Y

U W P

Compare this result to the values

Row 1 (R1)

3

5

1

0

0 78

obtained using Method I (graphical

Row 2 (R2)

4

1

0

1

0

9

method)
...


Row 3 (R3)

-5

-4

0

0

1

0

Then use row operation to make that 4 into
Simplex Method from YouTube:

1, so we’re going to replace Row 2 with the

(di binigay yung example)

formula of

Maximize 𝑃 = 5𝑥 + 4𝑦 ≤

1
4

𝑅2
1

(bali magiging 4 into 4 para maging 1)

3𝑥 + 5𝑦 ≤ 78

X

Y

U W P

Row 1 (R1)

3

5

1

0

0 78

Row 2 (R2)

4

1

0

1

0

9

Row 3 (R3)

-5

-4

0

0

1

0

4𝑥 + 𝑦 ≤ 36
𝑥 ,𝑦 ≥ 0
then do this: (yung letter u and w, kahit
anong letter na prefer mo)

(idi-divide lahat ng nasa Row 2 (X to P) ng

3𝑥 + 5𝑦 + 𝑢 = 78

1

4𝑥 + 𝑦 + 𝑤 = 36

4

but 9 is still the same so don’t divide it
...
25 = 12

= 9 ÷ 0
...
Then we need y column to replace
4

into 1 (bali magiging 17 into
1)

R1

11

11 4

+
4

P=1+

4

(0) = 0

4

11

P

W

11

11

X=0+

0 + 5(9) = 45

Row 3 (R3)

3

W = 17(− 4) = − 17

1
11
𝑌 = −4 + 5( ) = −
4
4

17
4

para maging

Now we can read off the answers, every
column that has a single 1 and two zeroes
are going to give us a variable:
X = 6 (6 sagot kasi yung position ng
numbers ay 0-1-0 tapos si 1 ay nasa middle

then kukuning sagot is yung 6 sa pinaka
dulo)
Y = 12 (12 sagot kasi yung position ng
numbers ay 1-0-0 tapos si 1 ay nasa top then
kukuning sagot is yung 12 sa pinaka dulo)
U = 78 (78 sagot kasi yung position ng
numbers ay 0-0-1 tapos si 1 ay nasa bottom
then kukuning sagot is yung 78 sa pinaka
dulo)
Tapos yung mga walang two zeroes and
single 1 ay considered as 0
Therefore:
U and W = 0



---