Matriks Matematika Kelas 11 • Part 23: Menyelesaikan SPLTV dengan Metode Determinan Matriks

Summary

This video from the 'Jendela Sains' channel delves into solving systems of linear equations with three variables using determinants. It explains the concept of determinants for 2x2 matrices and extends it to 3x3 matrices, essential for solving such systems. The tutorial walks through the steps of calculating determinants using the Sarrus' rule and applying them to find the values of variables in a system. It also includes a practical example involving a school library's collection of science, history, and religion books, demonstrating how to set up equations and solve them using determinants.

Takeaways

• 📚 The video discusses solving a system of linear equations with three variables using determinants, specifically focusing on the third-order determinants.
• 🔢 It explains that while two-variable linear systems can be solved using second-order determinants, three-variable systems require third-order determinants.
• 📐 The video introduces the concept of 'Delta', which is the determinant of a 3x3 matrix containing the coefficients of the variables x, y, and z from the equations.
• 📝 It demonstrates the method of calculating 'Delta X', 'Delta Y', and 'Delta Z' by removing the respective variable's coefficients and replacing them with constants from the right-hand side of the equations.
• 👨‍🏫 The tutorial uses the Sarrus' rule to calculate the determinants, which is a method for finding the determinant of a 3x3 matrix.
• 📘 An example problem is presented involving a school library with a collection of science, history, and religion books, aiming to find the number of each type of book.
• 🔄 The script details the process of setting up equations based on the problem statement and then solving for the variables using the determinant method.
• 🧮 The video shows step-by-step calculations for 'Delta', 'Delta X', 'Delta Y', and 'Delta Z', including the application of Sarrus' rule with the given numbers.
• 📊 It concludes with the solution to the example problem, determining the number of science, history, and religion books in the library.
• 💡 The video emphasizes the practical application of determinants in solving real-world problems, such as inventory management in the context of the example.

Q & A

What is the main topic discussed in the video?

The main topic discussed in the video is solving systems of linear equations with three variables using determinants of matrices.

What is the significance of determinants in solving systems of linear equations?

Determinants are used to find the unique solution of a system of linear equations. In the context of the video, they are used to solve a system of three variables.

What is the difference between solving a system of linear equations with two variables versus three variables?

In the video, it is mentioned that the principle of solving a system of linear equations with two variables is simpler compared to three variables, where determinants of a 3x3 matrix are used instead of a 2x2 matrix.

What are the steps to find the determinant of a 3x3 matrix using the method of Sarrus?

The steps include calculating the determinant by multiplying the elements of the main diagonal and summing them, then subtracting the sum of the products of the elements of the diagonals that are perpendicular to the main diagonal.

How does the video demonstrate the application of determinants to a real-world problem?

The video demonstrates the application of determinants by solving a problem involving a school library's collection of science, history, and religion books, where the number of each type of book is unknown.

What is the relationship between the number of science books and history books according to the problem presented in the video?

The relationship is given as a ratio of 5 to 8, meaning for every 5 science books, there are 8 history books.

How does the video handle the situation where the number of religion books is 100 more than the number of science books?

The video sets up an equation where the number of religion books (z) is represented as the number of science books (x) plus 100, and then solves for z using determinants.

What is the final outcome of the example problem presented in the video?

The final outcome is that there are 250 science books, 400 history books, and 350 religion books in the library.

What method does the video recommend for simplifying the process of solving the system of equations?

The video recommends using the method of Cramer's rule, which involves calculating the determinants of matrices with the coefficients of the variables and then finding the variables by dividing these determinants by the main determinant.

How does the video ensure that the viewers understand the process of solving the system of equations?

The video ensures understanding by walking through each step of the process, providing clear explanations, and demonstrating the method with a practical example.

Outlines

00:00

📚 Introduction to Solving Systems of Linear Equations with Matrices

This paragraph introduces the topic of the video, which is about solving systems of linear equations with three variables using determinants of matrices. The presenter explains that the process is similar to solving systems with two variables but involves a 3x3 matrix instead of a 2x2. The video aims to guide viewers through the process of solving such systems using determinants, starting with the first method, which involves finding the determinant of the matrix containing the coefficients of the variables x, y, and z from the given equations. The presenter also mentions the use of the Sarrus' rule for calculating determinants and provides a step-by-step approach to finding the values of x, y, and z.

05:00

🔍 Detailed Calculation Process Using Determinants

This paragraph delves into the detailed calculation process for solving the system of equations using determinants. It describes the method of finding the determinants for each variable (x, y, and z) by successively removing the coefficients of that variable and replacing them with the constants from the right side of the equations. The presenter uses the Sarrus' rule to calculate the determinants and provides a step-by-step guide on how to perform these calculations. The paragraph includes a practical example of a library collection problem, where the goal is to find the number of science, history, and religious books based on given ratios and total number of books.

10:04

📈 Final Calculations and Conclusion

The final paragraph wraps up the calculations for the library collection problem. It presents the final steps in calculating the determinants for each variable and then finding the values of x, y, and z. The presenter provides the final values for the number of science, history, and religious books in the library. The paragraph concludes with a summary of the video's content and an invitation for viewers to explore more topics in the playlist. It also encourages viewers to leave comments with questions, suggestions, or critiques.

Keywords

💡Matrix

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of the video, matrices are used to represent and solve systems of linear equations. The video discusses solving a system of three linear equations with three variables using a 3x3 matrix, which is a fundamental concept in linear algebra.

💡System of Linear Equations

A system of linear equations refers to a collection of two or more linear equations involving the same set of variables. The video specifically addresses systems with three variables, which are represented by three equations. These systems can be solved using matrices to find the values of the variables that satisfy all equations simultaneously.

💡Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It plays a crucial role in solving systems of linear equations, as it helps determine if a unique solution exists. In the video, the determinant of a 3x3 matrix is calculated to solve the system of equations, showcasing its application in finding the values of x, y, and z.

💡Sarrus' Rule

Sarrus' Rule is a method for calculating the determinant of a 3x3 matrix. The video demonstrates this rule by applying it to find the determinants needed to solve for the variables in the system of equations. It's an efficient way to compute determinants without expanding the matrix.

💡Variables

In the context of the video, variables refer to the unknowns in the system of linear equations, represented by x, y, and z. The goal of solving the system is to find the values of these variables that satisfy all the equations. The script uses these variables to form the equations and demonstrates how to isolate and solve for each one.

💡Coefficients

Coefficients are the numerical factors that multiply the variables in the equations. In the video, coefficients are the numbers that precede the variables x, y, and z in each equation. They are crucial for setting up the matrix and are used in the determinant calculations to solve for the variables.

💡Constants

Constants are the numerical values on the right side of the equations in a system of linear equations. They represent the results that the left side of the equations should equal when the variables are correctly solved for. In the video, constants are used to form the equations and are part of the determinant calculations for solving the system.

💡Delta (Δ)

In the context of the video, 'Delta' refers to the determinants calculated from the original matrix with certain columns replaced by the constants from the equations. These determinants, denoted as Δx, Δy, and Δz, are used to find the values of x, y, and z. The script demonstrates how to calculate these determinants using the Sarrus' Rule.

💡Method of Elimination

Although not explicitly mentioned in the transcript, the method of elimination is implied in the process of solving the system of equations. It involves manipulating the equations to isolate and solve for one variable at a time, often by using determinants and matrices. The video uses determinants to eliminate variables and find unique solutions.

💡Example Problem

The video includes an example problem involving a school library's collection of science, history, and religion books. This practical application demonstrates how matrices and determinants can be used to solve real-world problems, specifically in this case, to find the number of books in each category.

Highlights

Introduction to solving a system of linear equations with three variables using determinants.

Explanation of the principle behind solving a system of linear equations with determinants.

The difference between solving a system of two variables and three variables using determinants.

How to use a 3x3 determinant matrix for a system with three variables.

The process of setting up the determinant matrix with coefficients from the system of equations.

The method to find the determinant of a 3x3 matrix using the Sarrus' rule.

How to calculate Delta X by removing the X coefficients and replacing them with constants from the right side of the equations.

The process of calculating Delta Y and Delta Z by removing Y and Z coefficients respectively.

The final step of finding the values of x, y, and z by dividing the respective Delta values by the main determinant.

A practical example involving a school library's book collection to illustrate the method.

Setting up the system of equations based on the book collection example.

Using the determinant method to solve for the number of science, history, and religion books in the library.

The step-by-step calculation of the determinant for the book collection example.

The final solution for the number of books in each category using determinants.

Encouragement for viewers to watch the complete playlist for more detailed information.

Invitation for viewers to leave comments with questions, suggestions, or critiques.

Closing remarks and a tease for the next video in the series.

Transcripts

Hai

semua

selamat datang di channel jendela sains

di video ini kita akan membahas matriks

part yang ke-23 yaitu tentang

menyelesaikan sistem persamaan linear

tiga variabel atau spltv dengan

determinan matriks

simak terus video ini sampai akhir

prinsip menyelesaikan sistem persamaan

linear tiga variabel dengan determinan

matriks itu sama dengan waktu di sistem

persamaan linear dua variabel cuma waktu

di SPLDV kan kita gunakan determinan

matriks ordo 2 * dua sekarang kita

gunakan determinan matriks ordo 3 kali 3

karena variabelnya ada tiga Oke jadi

kalau kita punya spltv dan pasti ada

tiga bersamaan ya Persamaan pertama masa

tu X + B satu y + z = b satu persamaan

kedua Aduh Express duaji + 2z = di dua

persamaan ketiga A3 x + 3 y + 3z = 3

maka A1 B1 c1di 1A 2B 2C 2D 2/3 b 3 C 3

D 3 ini kan semuanya sudah diketahui ya

Jadi kita disuruh mencari berapa

xy&z maka caranya seperti ini kita

gunakan cara determinan yang pertama

adalah kita cari delta-delta itu apa

Delta itu adalah determinan dari matriks

ordo 3 kali 3 yang berisi koefisien xyz

di setiap persamaan yakni kan sampai A1

B1 J1 A2 B2 c-2a 3B 3C tidak sama

seperti ini Kita cari determinannya ini

dengan metode sarrus Oke berikutnya kita

cari Delta X kalau data itu adalah

determinan dari matriks matriks nya

Delta cuma x-nya koefisien X itu dihapus

jadi koefisien isn't A1 A2 A3 dihapus

ini posisinya digantikan dengan

konstanta yang diruas kanan ini D1 D2 D3

jadinya matriksnya seperti ini dicari

determinannya sama dengan metode sarrus

berapa ketemunya berikutnya delta j-talk

delta ye berarti dari Delta ini

koefisiennya dihapus tadi ini Tengah ini

B1 B2 B3 ya kan dihapus digantikan

dengan konstanta yang ada di ruas kanan

lebih jadi D1 D2 D3 di tengah Oke

dihitung lagi dengan metode sarrus

berapa determinannya yang terakhir Delta

z sama berarti koefisiennya zc1 c233

pada Delta ini digali dengan D1 D2 D3

berarti matriksnya seperti ini tinggal

cari determinannya Oke bagian ketemu nih

hasil ekstrak dari Delta delta X dan

tadi dan dataset lalu tinggal kita cari

xy&z Hai X = Delta ekspor Delta y =

Delta y perdata dan Z = Delta Z per

Delta Oke untuk lebih memudahkan

memahami kita langsung ke contoh soal

sebuah perpustakaan sekolah mengoleksi

1000 eksemplar buku bacaan yang terdiri

dari buku sains buku sejarah dan buku

Agama perbandingan banyak buku sains dan

buku Sejarah adalah 5 banding 8

sedangkan buku agama 100 eksemplar lebih

banyak dibandingkan buku sains dengan

menggunakan determinan matriks Tentukan

banyak buku sains buku sejarah dan buku

agama masing-masing yang ada

diperpustakaan Hidup berarti permisalan

nya tips itu adalah banyak buku sains

lalu dia itu adalah banyak buku sejarah

dan z Itu adalah banyak buku agama

kita buat persamaannya ya disitu 1000

eksemplar itu terdiri dari buku sains

buku sejarah dan buku Agama ya pasti

langsung bisa kita tulis x + y + z Itu

sama dengan

1000D ini persamaan 1 lalu perbandingan

banyak buku sains seperti X dan buku

sejarah tadi X banding Y atau kita buat

pecahan aja explore y = 5 atau delapan

kita Kali Bilang aja berarti 8X = 5 y

Rim akhirnya kita pindah ke kiri berarti

8 x min 5 y = 0 lalu buku agama Raffi

z100 eksemplar lebih banyak dibandingkan

buku sains berarti X plus 100

berarti esnya Kita pindah ke kiri min x

+ z = 100% aman tiga ya Yang ini tadi

persamaan

oke lalu gini Agar tidak membingungkan

kita poles ulang ini tiga persamaan ini

yang pertama x + y + z =

1221 x min 5 y enggak ada seinnya ya

tapi di sini tips dari saya ditulis 0-z

plus 01 = No biar nggak bingung nanti

waktu hitung delta delta XL tajwid dan

detached yang terakhir bersama ketiga

minex hanya enggak ada berarti plus 0

+ Z =

111 ya Sekarang kita mulai dari hitung

Delta tidak tahu itu berarti determinan

dari ya koefisien x y z dari persamaan

123 ya batin

111 lalu persamaan ke-28 Min 50

bersamaan ketiga bertines

1001 Oke kita gunakan metode sarrus kita

salin kolom pertama tadi

0851 dan kolom ke-21 Min 50 lalu di sini

kita buat Garis yang sejajar diagonal

utama tandanya plus plus plus dan Garis

yang sejajar diagonal samping ini

berarti tandanya ini minus

Oke kita mulai cari Delta berarti satu

kali mi5 kali satu berarti kan Mini Maya

plus satu kali nol Kalimin satu berarti

0 plus satu kali 8000 juga minus

sekarang mint satu kali menerima kali

satu berarti lima

clean 001 pasti 0 min satu kali delapan

kali satu bagi

83 ini = Min 5 Min 5 Min 10 Min 8

berarti Min 18 berikutnya kita cari

Delta X

j&t Express di Sekarang koefisiennya

x181 kita ganti dengan ini Akon santai

diruas kanan 1000 0-100 lebih jadi

1000.00 kini 100 dan tengah sama

koefisiennya tetap 1min

500 koefisien shootnya

101 tak lu kita salin kolom pertama sama

kolom kedua lalu kita bergaris dan tanda

kalau sudah kita hitung

seribu kali min 5 kali satu berarti mint

Rp5.000

plus satu kali 0-100 pasti 0 plus satu

kali 000 Min 100 K5 kali satu berarti

Mini Mar'atus ya atau mint kelamin

langsung aja plus500

mint nol kali nol kali 1000 Yamin nol

lalu mint satu kali 01 yang menolak

Hai = Min 5 ribu plus500 bagi mint 4500

Oke sekarang kita cari Delta

y-delta ye berarti dari Delta ini

koefisiennya yang Tengah 1 Min 50 ini

diganti dengan ini konstanta diruas

kanan

1600 koefisien X yang tetap

18 mint 11 ini 1000

0-100 office nya tetap

101 Oke kita salin kolom pertama sama

kolom kedua

kalau sudah kita kasih garis-garis dan

anda

kalau sudah kita hitung batin sama

dengan yang pertama satu kali nol kali

10 blouse 1000 kali nol Kalimin 10 plus

satu kali delapan kali 100 berarti

halo halo minus minus satu kali 010

minus lagi 100 Thailand 010

minus satu kali delapan kali

1038 ribu

hati sama dengan 800 mil 8000 per

diminum 7200

berikutnya kita cari Delta Z Delta set

ini berarti koefisien x nya tetap

18 min 1 koefisien Joe tetap 1 Min 50

koefisiensi at-101 digantikan dengan

konstanta

1600

[Musik]

oke lalu kita salin kolom pertama sama

kolom kedua

oke kalau sudah kita buat garis-garis

dan tandanya

Oke kalau sudah kita hitung lebih sama

dengan yang pertama satu kali min 5 kali

min 100 berarti Min 500

blouse satu kali Gimin 10 Plus 1000 kali

8000

minus minus 1 Kalimin 55 kali 1000-5000

saat ini 5000

Min 00 kali 10 Min 100 kali delapan kali

satu peti

800ht = Min 505000 kandeman

5587 batin 6300 Oke kalau sudah kita

cari xy&z hex babi = Delta expert Delta

berarti = Delta Iskan ini ya Min 4500

perdatanya Min 18 Min 4 tipe 500 dibagi

minus 18 berarti hasilnya 250 lalu y =

Delta y per Delta berarti Delta yekan

ini ya Min

terus

permen 18 Bakti hasilnya 400

dan Z = Delta Z for Delta ngerti sama

dengan ini

6300

dibagi sama Min 18 hasilnya adalah

350 Oke jangan lupa satuannya eksemplar

per ya Oke berarti kita dapatkan

jawabannya banyak buku sains itu 250

eksemplar banyak buku Sejarah itu 400

eksemplar dan banyak buku agama 350

eksemplar

oke sekian untuk video kali ini untuk

melihat playlist lengkap dari bab ini

bisa kalian Klik tombol yang ada di

sebelah kanan atas ini jika ada

pertanyaan saran maupun kritik bisa

kalian tulis di kolom komentar semoga

bermanfaat dan sampai jumpa divideo

selanjutnya dadaa