Matriks Matematika Kelas 11 • Part 23: Menyelesaikan SPLTV dengan Metode Determinan Matriks
Summary
TLDRThis 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
📚 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.
🔍 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.
📈 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.
Mindmap
Keywords
💡Matrix
💡System of Linear Equations
💡Determinant
💡Sarrus' Rule
💡Variables
💡Coefficients
💡Constants
💡Delta (Δ)
💡Method of Elimination
💡Example Problem
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
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