Program Linear (Part 1) Pertidaksamaan Linear Dua Variabel

Summary

This educational video script introduces linear programming for high school students, focusing on two-variable linear inequalities. It explains the general form of such inequalities and demonstrates how to graph them by finding intercepts and connecting points. The script also guides viewers on determining the solution region by testing points against the inequality, emphasizing the importance of identifying the correct area that satisfies the condition. The lesson aims to equip students with the skills to solve practical problems related to linear programming in various fields.

Takeaways

• 📚 The lesson is focused on teaching linear programming for 11th-grade high school students, emphasizing its importance in various fields such as the garment industry and trade.
• 🔍 The script introduces linear equations with two variables, building upon the understanding of linear equations with one variable from the 10th grade.
• 📈 The general form of a linear equation with two variables is presented as 'Ax + By = C', where the sign can vary, including 'greater than', 'less than', or 'equal to'.
• 📝 The process of finding the solution set for a linear equation with two variables involves graphing the equation and identifying the intersection points with the coordinate axes.
• 📍 To graph a line, the script explains how to find the x-intercept by setting y to zero and solving for x, and the y-intercept by setting x to zero and solving for y.
• 📉 The script provides a step-by-step method to determine the solution area of a linear equation, starting with graphing the line and then testing points to see if they satisfy the inequality.
• 🚫 The importance of correctly identifying the solution area is stressed, with the script explaining how to eliminate areas that do not satisfy the inequality.
• 📐 The script uses examples to illustrate the process, including how to handle different types of inequalities and how to interpret the results of the tests.
• 🔎 The method of testing a point, such as the origin (0,0), is highlighted to determine whether it lies above or below the line, which helps in identifying the correct solution area.
• ✂️ The script emphasizes the practical application of linear programming, suggesting that the students should practice at home to better understand the concepts.
• 🌟 The lesson concludes with a reminder of the usefulness of linear programming and a wish for the students to benefit from the knowledge, ending with a respectful sign-off.

Q & A

What is the main topic of the video script?

The main topic of the video script is the study of linear programming for 11th-grade high school mathematics, focusing on linear inequalities with two variables.

What is the significance of learning linear programming in various fields?

Linear programming is significant in various fields such as the garment industry, trade, and others because it is widely applicable in solving real-life problems.

What is the general form of a linear inequality with two variables?

The general form of a linear inequality with two variables is \( Ax + By \neq C \), where \( A \), \( B \), and \( C \) are constants, and the inequality sign can be 'not equal to', 'greater than', 'less than', 'greater than or equal to', 'less than or equal to'.

How do you find the intersection points of a line with the x-axis?

To find the intersection point with the x-axis, set \( y = 0 \) and solve for \( x \) from the equation of the line.

How do you find the intersection points of a line with the y-axis?

To find the intersection point with the y-axis, set \( x = 0 \) and solve for \( y \) from the equation of the line.

What is the first step in solving a system of linear inequalities?

The first step in solving a system of linear inequalities is to graph the lines representing each inequality on a coordinate plane.

How do you determine the solution region for a linear inequality?

The solution region is determined by testing a point, such as the origin (0,0), in the inequalities to see if it satisfies the condition of the inequality.

Why is it important to identify the area that is not part of the solution set?

Identifying the area that is not part of the solution set helps in simplifying the process of finding the actual solution region and avoids confusion when dealing with multiple inequalities.

What is the process of eliminating non-solution areas in the context of the script?

The process involves testing points in the inequalities and eliminating the areas that do not satisfy the inequality conditions, thus narrowing down to the actual solution region.

Can you provide an example of a linear inequality with a negative inequality sign as mentioned in the script?

An example of a linear inequality with a negative inequality sign is \( 4x - 3y \leq 12 \), which represents a situation where the value of the expression on the left must be less than or equal to 12.

What does the script suggest for the final step in determining the solution region of a system of linear inequalities?

The final step suggested in the script is to eliminate the areas that do not satisfy the inequalities, leaving behind the actual solution region where all conditions are met.

Outlines

00:00

📚 Introduction to Linear Programming in Mathematics

This paragraph introduces the topic of linear programming, a subject essential for 11th-grade high school mathematics. It highlights the practical applications of linear programming in various fields such as the garment industry, trade, and more. The speaker explains the importance of understanding linear equations with two variables, which are foundational for solving linear programming problems. The paragraph also covers the general form of a linear equation with two variables, represented as 'Ax + By = C', where 'A', 'B', and 'C' are constants, and 'x' and 'y' are the variables. The speaker clarifies that the equality sign can be replaced with other relational operators to represent different types of inequalities.

05:00

📈 Graphical Method for Solving Linear Equations

The second paragraph delves into the graphical method for solving linear equations with two variables. It begins by explaining how to plot the line represented by the equation '2x + 3y = 6'. The process involves finding the x-intercept by setting 'y' to zero and solving for 'x', resulting in an x-intercept of (3, 0). Similarly, the y-intercept is found by setting 'x' to zero, leading to a y-intercept of (0, 2). The speaker then illustrates how to connect these intercepts with a straight line. The next step is to determine the solution area by testing a point, such as the origin (0, 0), in the inequality '2x + 3y ≥ 6'. The point (0, 0) does not satisfy the inequality, indicating that the solution area is above the line.

10:03

📉 Advanced Graphical Techniques for Linear Inequalities

This paragraph continues the discussion on solving linear inequalities with two variables but introduces a more efficient method by using a 'box' to quickly find the x and y intercepts. The example given is the inequality '3x + 5y ≤ 15'. The speaker demonstrates how to calculate the intercepts by setting 'y' to zero to find 'x' and vice versa, resulting in intercepts at (5, 0) and (0, 3). The intercepts are then connected to form a line, and the solution area is determined by testing the point (0, 0), which lies below the line, confirming that the area below the line is the solution region for the inequality.

15:05

🔍 Determining Solution Areas for Complex Linear Inequalities

The final paragraph addresses how to determine the solution area for a linear inequality with a negative coefficient, using the example '4x - 3y ≤ 12'. The process involves plotting the line and finding the intercepts, which are (3, 0) and (0, -4) for this particular inequality. The speaker then explains the testing process using the point (0, 0) and confirms that it satisfies the inequality, indicating that the solution area is to the right and below the line. The paragraph emphasizes the importance of correctly identifying which side of the line represents the solution area and the practicality of this method when dealing with systems of linear equations.

Keywords

💡Linear Program

A linear program is a mathematical optimization technique that deals with linear relationships between variables. In the context of the video, it is a fundamental concept for solving real-world problems in industries such as manufacturing and trade. The script discusses the importance of understanding linear programs for their wide range of applications.

💡Linear Inequality

A linear inequality is a mathematical statement where the expression on the left side of the inequality sign is not equal to the expression on the right side. The video script explains how to solve linear inequalities with two variables, which are essential for understanding the broader concept of linear programming.

💡Variables

In mathematics, variables are symbols that represent values that can change. The script introduces the concept of variables in the context of linear inequalities, where 'x' and 'y' are used to represent the unknown quantities that need to be determined.

💡Graphical Representation

Graphical representation is a method of visually depicting data or mathematical relationships. The video script describes how to graphically represent linear inequalities by plotting lines and points to find the solution set.

💡Intersection Points

Intersection points are the points where two lines or curves meet on a graph. In the script, finding the intersection points with the x-axis and y-axis is a crucial step in graphing linear inequalities and determining the solution area.

💡Solution Set

The solution set in the context of inequalities refers to the set of all possible values that satisfy the inequality. The video script explains how to determine the solution set by graphing and analyzing the regions that satisfy the given linear inequalities.

💡Axis

In a Cartesian coordinate system, the axis refers to the lines that intersect at the origin and serve as a reference for all other points. The script uses the terms 'x-axis' and 'y-axis' to describe where to find the intersection points for graphing linear inequalities.

💡Inequality Sign

An inequality sign is a mathematical symbol that indicates that the relationship between two expressions is not one of equality. The script mentions different inequality signs such as 'greater than or equal to' and 'less than or equal to' in the context of linear inequalities.

💡Verification

Verification in mathematics is the process of checking whether a particular solution or set of solutions satisfies an equation or inequality. The video script describes how to verify the solution set by testing points within the graphed region to ensure they meet the inequality's conditions.

💡Coordinate System

A coordinate system is a reference frame for graphing where each point is defined by a pair of numerical coordinates. The script uses a coordinate system to plot the points and lines necessary for solving linear inequalities.

💡Solving Linear Inequalities

Solving linear inequalities involves finding all the values of the variables that make the inequality true. The script provides a step-by-step method for solving these inequalities, which is central to the video's educational content.

Highlights

Introduction to linear programming for 11th-grade high school mathematics, a mandatory subject with wide applications in industries such as garment manufacturing and trade.

Explanation of linear programming as a fundamental concept to solve various real-life problems.

Transition from understanding linear equations with one variable to those with two variables.

General form of a two-variable linear equation is introduced, with examples of different inequality signs.

Methodology to find the solution set for two-variable linear equations by graphing.

Step-by-step guide on how to graph a line given an equation, starting with finding the intercepts.

How to calculate the x-intercept by setting y to zero and solving for x.

Similar process to find the y-intercept by setting x to zero and solving for y.

Connecting the intercepts with a straight line to visualize the equation graphically.

Verification process to determine the solution area by testing points above or below the line.

Using the point (0,0) for simplicity in testing whether it satisfies the equation.

Explanation of the implications of the test result on the solution area being above or below the line.

Strategy to discard the area not satisfying the equation to narrow down the solution area.

Demonstration of solving a specific linear inequality with steps to graph and verify the solution area.

Efficient method of using a box to find intercepts for graphing complex linear equations.

How to interpret the inequality sign in the equation to determine the correct area for the solution.

Application of the process to another example with a different linear equation and inequality sign.

Final summary of the method to determine the solution area for two-variable linear equations.

Encouragement for students to practice the concepts at home for better understanding.

Transcripts

Hai kewajiban penyebab putus

assalamualaikum warahmatullah

wabarakatuh semoga lagi dengan tentara

online kali ini kita akan belajar

tentang program linear kelas 11 SMA

wajib ya untuk Matematika wajib untuk

program linear merupakan salah satu

materi yang banyak penerapannya dalam

kehidupan ya bisa industri konveksi

pertokoan perdagangan dan lain

sebagainya sebagai dasar untuk bisa

menyelesaikan masalah program linier

kita akan belajar tentang yang pertama

adalah pertidaksamaan linear dua

variabel kita sudah mengenal waktu kelas

10 mungkin ya pertidaksamaan linear satu

variabel misalkan X

Hai dengan x kurang dari 5 Misalkan

begini atau Y lebih dari tujuh ini ini

namanya pertidaksamaan linear satu

variabel Sekarang kita akan belajar

pertidaksamaan linier dua variabel saya

kita aku saja ini bagaimana bentuk umum

dari pertidaksamaan linear dua variabel

bentuk umumnya seperti ini A X plus b y

nah tanda disini bukan sama dengan ya

Tapi bisa juga mungkin lebih dari = C

seperti itu tanda ini bisa diganti

dengan tangan yang lain ya bisa mungkin

lebih dari saja atau mungkin kurang dari

atau sama dengan atau mungkin kurang

dari saja

inilah bentuk umum dari tertindas teman

linier dua variabel kemudian kita akan

diminta untuk mencari himpunan

penyelesaian dari pertidaksamaan linear

dua variabel ini bagaimana caranya saya

ambil contoh ya contoh Tentukan daerah

penyelesaian dari 2x + 3 Y lebih dari

atau sama dengan enam langkah pertama

kita harus menggambar garis 2x y + 3 Y =

6 yang pertama gambar garis 2 x + 3y =

ia masih ingat Bagaimana cara menggambar

Paris seperti ini ya yang pertama adalah

cara menggambar garis ya yang pertama

[Musik]

mencari titik potong titik potong

terhadap sumbu x sumbu x berada di mana

di sini ya nih Apa syaratnya Agar sebuah

garis memotong sumbu x seratnya adalah

nilai Heroes lol ini diperoleh ketika y

= 0 kita akan menghitung nih Berapa

nilai x nya kalau nya sama dengan nol

tadi diperoleh 2x + karyanya katinon

tiga kali 0 = 6

yang diperoleh 2x = 6 tahu x = 3 jadi

titik potong terhadap sumbu x adalah 0,3

atau 30 ya jangan salah ya 3,0 bukan 0,3

ini jangan sampai kebalik jadi titik

potongnya titik potongnya 3,0 hati-hati

bukan 0,3 yang kedua mencari titik

potong terhadap sumbu y mencari titik

potong titik potong terhadap sumbu y

Hai Apa syaratnya syaratnya adalah x = 0

ya kita ganti setengah nonprotein dua

Kalian 0x disini kenal ya kita Mbah 3y =

nah diperoleh 3y = 6 tahu Y = 2 ya jadi

titik potongnya adalah makhluk 4242

konon ya 0,2 ya X terlebih dahulu jadi

titik potongnya titik potongnya adalah

0,2 kita masukkan dalam kabar ya yang

pertama 30123 Disini

[Musik]

Hai bukan di sini ya tapi di sini

kemudian 0,2 berada di sumbu y di sini

berarti sini gua selanjutnya kita

hubungannya kedua titik potongnya dengan

sebuah garis lurus ya seperti ini

setelah garis yang terbentuk langkah

berikutnya adalah kita harus menguji

data mengambil titik diatas atau dibawah

garis ini ya ketika memenuhi berarti

daerah itu telah daerah penyelesaian

Bagaimana cara mengujinya kita milik

Semarang bisa diatas bisa di bawah tapi

untuk memudahkan pengujian kita kita

ambil titik yang paling mudah yaitu

titik 0,0 titik 0,0 berarti terletak

dibawah garis ini Langkah kedua adalah

uji titik

[Musik]

Hai MUI memohon Wow 0,0 dimasukkan

kedalam pertidaksamaan ini Baik + 3 Y

lebih dari = 6 Rt xh330 iyanya diganti

nol tapi dari enam diperoleh non lebih

dari enam ini pernyataan ini benar atau

salah apakah lebih dari enam nah Berarti

pernyataan ini tentu saja salah ya Salah

[Musik]

Kalau salah berarti daerah yang sepihak

dengan titik nol ini artinya yang berada

dibawah garis ini bukan daerah

penyelesaian daerah penyelesaiannya

berat

di atas saat ini kita akan mengarsir

yang bukan daerah ingat yang diarsir

yang bukan daerah penyelesaian jadi yang

bersih nanti malah yang daerah

penyelesaian kenapa saya memilih itu

karena nanti ketika kita sudah sampai

sistem pertidaksamaan linier akan banyak

sekali arsiran sehingga akan kesulitan

mencari daerah penyelesaian jika yang

diarsir adalah yang daerah hasil

terakhir kebawahin ya Nis diarsir

pokoknya bukan daerah pilihan di daerah

penyelesaian senjata penyelesaian yang

bawa malah bukan bukan daerah

penyelesaian seperti itu

Hai cara menentukan daerah penyelesaian

dari pertidaksamaan linear dua variabel

kita akan membahas contoh yang kedua ya

di sini sudah ada pertanyaan Tentukan

daerah penyelesaian dari pertidaksamaan

linear 3x + 5 Y kurang dari sama dengan

15 langkahnya seperti tadi yang pertama

adalah membuat gambar-gambar gambar

garis 3 x + 5y =

Hai caranya tadi adalah mencari titik

potong terhadap sumbu x dan sumbu y

untuk selanjutnya kita percepat langkah

menggambar ini kita akan gerhana kan

caranya kita buat kotak seperti ini ya

kita buat kotak dengan posisi seperti

ini yang ini adalah untuk mencari titik

potong terhadap sumbu x dimana nilai y =

0 Perhatikan Kalau ini nol berarti 3x =

15 artinya x = 5 jadi perhitungan ini

sudah di Iran kita ya ini lima kemudian

yang ini adalah titik potong

langsung piye gimana x = 0 ketika

excalor artinya apa 5y = 15 berarti y =

3 diperoleh titik potong terhadap sumbu

x 5,0 terhadap sumbu-y 0,3 ini 5,0 yang

ini 4,3 kemudian seperti tadi hubungkan

ya kedua titik tersebut Ya seperti ini

Eh ini mohon maaf ya kurang pas ya tapi

intinya seperti ini berikutnya adalah

Kita uji titik O titik apa 0,0 uji it

klik 0,0 kita masukkan 0,0 dalam

pertidaksamaan yang ini tiga kali non

tambah lima kali 03 kalian 00 ini lima

kali no juga nonprotein Oh kurang dari

15 bener nggak nol kurang dari 15 benar

ya Nah karena benar maka semua

daerah-daerah aslinya adalah daerah yang

berada di titik 0,0 ini ya artinya

daerah yang dibawah garis ini merupakan

daerah penyelesaian dari pertidaksamaan

linear 3x + 5 Y kurang dari sama dengan

15 yang atas bukan yang daerahnya di

bawah yang atas bukan daerah

penyelesaian

Hai yang diarsir Sekali lagi saya

tekankan yang bukan daerah jadi kasirnya

ini atasmu ya Ini bukan ya bukan daerah

penyelesaian yang ini daerah

penyelesaian semuanya seperti itu ya

saya akan berikan satu contoh lagi

Bagaimana kalau misalkan tandanya teh

negatif seperti apa ya kita akan

membahas contoh yang ketiga

pertidaksamaan linear 4x min 3 Y kurang

dari atau sama dengan 12 Nah pertama

bersama ya gambar garis

ke-4 X min 3 Y = 12 membuat kotak

seperti ini Nah untuk mencari titik

potong terhadap sumbu x dan sumbu y

Hai mungkin Allen kamu ini xy017 titik

potong terhadap sumbu x y nya non ketika

ia nya norma ke-4 = 12 detik detik Aya

Diga ketika Excel long rotimi negatif 3y

= 12 artinya y = negatif 4 jadi titik

potongnya 3,0 sini kemudian lol

komunikatif 40234 sini negatif 4

selanjutnya kita hubungkan kedua detik

ini ya Ya seperti ini selanjutnya tangga

kedua adalah menguji

Hai tanker laut titik non uji titik 0,0

kita masukkan ke pertidaksamaan ini

matte X min 3 Y kurang dari sama dengan

empat kali North dikurangi tiga kali nol

kurang dari 1300 kurang dari 12 benar

atau salah benar kalau Benar berarti

daerah penyelesaiannya adalah yang

sepihak dengan long seperti berada di

sini yang yang sebelah kanannya berarti

bukan daerah penyelesaian ingat

mengarsirnya kemana ke yang bukan ya

betina artinya kesini kan Bawa charger

semuanya terus sampai ya ini telah

bukan-bukan

Hai Tera penyelesaian yang sebelah sini

semuanya ini adalah daerah penyelesaian

ya demikian pembahasan kita tentang

pertidaksamaan linear dua variabel

silahkan kalian pelajari di rumah

Selamat mencoba Selamat belajar semoga

bermanfaat saya akhirnya selama alaikum

warahmatullah wabarakatuh Pak