Systems of Linear Equations and Solving Them Using Cramer’s Rule
Kokand State University
Faculty of Exact Sciences and Digital Technologies
Student: Mamadaliyeva Durdona
E-mail:durdonamamadaliyeva60@gmail.com
Abstract: In this scientific article, the theoretical foundations and practical application of solving systems of linear equations using Cramer’s rule are described. The method is based on determining the unknowns through determinants and is applicable only when the main determinant is nonzero. The algorithm steps are demonstrated on examples with two and three unknowns, and the general form of the formulas is presented.
Keywords: Cramer’s rule, trapezoid, system of linear equations, algebraic method, determinant, constant term.
Аннотация: В данной научной статье раскрываются теоретические основы и практическое применение решения систем линейных уравнений методом Крамера. Метод основан на определении неизвестных с помощью определителей и применяется только в случаях, когда главный определитель не равен нулю. На примерах с двумя и тремя неизвестными показаны этапы алгоритма и приведена общая форма формул.
Ключевые слова: метод Крамера, трапеция, система линейных уравнений, алгебраический метод, определитель, свободный член.
Systems of linear equations arise in many fields of science, engineering, and economics. Solving such systems efficiently and accurately is an important skill in linear algebra. One of the classical methods for solving systems with a unique solution is Cramer’s Rule, which is based on the concept of determinants.
For systems with a non-zero determinant, Cramer’s rule is one of the simplest and most convenient methods to understand and apply.
This article examines the strategic role of globalization, innovation, and investment in ensuring the sustainable development of entrepreneurial activity. In particular, globalization is seen as a driving force in the introduction of new technologies and the achievement of environmental sustainability, while investment serves as the key instrument for implementing these processes. The study analyzes the experiences of Uzbekistan and other countries, offering modern approaches and recommendations aimed at sustainable economic growth. Furthermore, it considers the role of globalization, innovation, and investment in the sustainable development of entrepreneurship, their interrelation, and effectiveness, using Uzbekistan as a case study. Based on international experiences, the paper explores crucial strategies and directions for contemporary entrepreneurship. These findings are of practical importance not only for business entities but also for institutions engaged in economic policy-making.
Keywords: Innovative economy, digital technologies, economic reforms, opportunities, transformation, global cooperation, environmental security.
Introduction
Today, sustainable development in the global economy has become a key objective not only for state policy but also for entrepreneurial activity. Innovations help address socio-economic challenges through technological solutions, while investments provide the financial basis for implementing these solutions. “Globalization involves economic and industry integration with the rest of the world, removing restrictions on imports and foreign investment… Globalization has created a knowledge-intensive economy, making firms’ search for the foreign market opportunities necessary in order to survive.”
Therefore, the sustainable development of entrepreneurs remains one of the key topics in the modern economy. For entrepreneurs, innovation is not merely technological advancement but also new approaches and solutions that lead to economic efficiency and social sustainability. Through innovation, it is possible not only to improve the quality of products and services but also to create new markets and opportunities. At the same time, investment serves as the financial foundation for the development of entrepreneurial activity. “Globalization boosts the process of market predictions in the pursuit of economic growth. When economic, political, and social forces gain traction by promoting policies and the climate favoring globalization, entrepreneurial results reach out.”
They provide the resources necessary to support innovation, enhance competitiveness, and meet market demands.
Analysis of the Literature Used
Scientific literature and research on sustainable entrepreneurship, innovation, and investment are important sources for gaining an in-depth understanding of these areas and developing effective strategies based on them. The reviewed literature can be thematically divided into the following groups:
Theoretical Research
Theoretical research focuses on studying the essence of economic development, innovation, and investment. For example:
As Schumpeter (1942) emphasized in his work “Capitalism, Socialism, and Democracy, innovation is regarded as a “creative destruction” mechanism of economic development.” It contributes to increasing competition and producing high-quality goods in the market by creating new technologies and products.
Porter and Kramer (2011): This study highlights the important role of innovation in addressing social and economic problems. The authors put forward the concept of “creating shared value,” stressing that directing business activities toward social issues can lead to more effective results.
Practical Research
Practical research examines the effectiveness of innovation and investment based on real-life examples and experiences. Studies in this group include the following:
McKinsey (2022): The company’s report analyzed digital technologies as a means of improving efficiency in entrepreneurship. The report noted the importance of automated systems, artificial intelligence, and digital services in business development.
OECD (2023): The report focused on the role of small and medium-sized enterprises (SMEs) in innovative development. Recommendations were provided on promoting sustainable economic growth through supporting SMEs.
Methodology
This study examines a number of similarities and differences between Uzbekistan and Germany—one of the world’s developed countries—in terms of entrepreneurship financing. Therefore, the use of a Venn diagram in this article is considered one of the most effective methods. The Venn diagram makes it possible to clearly illustrate the common and distinct aspects of the policies and practices of Uzbekistan and Germany in financing entrepreneurship.
Analysis and Results
Figure 1. Venn diagram
The results of the Venn diagram indicate that in both countries, defining entrepreneurship as a priority in state policy, establishing mechanisms to support innovative projects, and offering tax incentives contribute to improving the business environment. In Germany, the diversity of funding sources and the effectiveness of the dual education system, and in Uzbekistan, reliance on technology parks and foreign investment, define each country’s unique strengths and limitations.
Conclusion and Recommendations
Innovation and investment play a central role in the sustainable development of entrepreneurial activity. Innovation enables the introduction of new technologies, products, and services that serve economic and social sustainability. They are essential tools for increasing entrepreneurs’ competitiveness, exploring new markets, and achieving growth without harming the environment. Investment, by financing innovation, ensures the successful implementation of these processes. The experience of Uzbekistan demonstrates that state policies and strategies play a vital role in achieving sustainable development goals. Initiatives such as the Innovation Development Strategy, IT Parks, and free economic zones are creating significant opportunities for business entities.
Based on this analysis, the following recommendations can be proposed:
Workforce Development: Strengthen the focus on innovative knowledge and skills in the education system to train qualified specialists.
Expand International Cooperation: Adopt international experience in innovation and investment, and strengthen relations with foreign investors.
Advance Digital Transformation: Promote the widespread adoption of automated systems, artificial intelligence, and digital services.
Furthermore, the objective of this study is to ensure that innovative ideas go hand in hand with their practical implementation.
Example:
Online Business Simulator – Creating an online simulator for entrepreneurs. Through this interactive platform, users can experience all stages of business development—from starting their enterprise to producing products, and from developing a marketing strategy to bringing it to market. In the simulator, various challenges and opportunities arise, and users learn by choosing how to respond to them.
The principle of operation of the proposed interactive platform is as follows: the user starts a business in the simulator, chooses a product or service, develops a financial plan, and implements a marketing strategy. Each decision affects the user’s success or failure. This provides an opportunity to explore the nuances of starting a business and gain practical skills.
In conclusion, in the context of Uzbekistan, the combination of innovation and investment is a crucial factor not only in ensuring economic growth but also in increasing global competitiveness and achieving sustainable development. To successfully implement this process, it is necessary to strengthen cooperation between the state, entrepreneurs, and research institutions.
References
Abdullajonov, D., & Mamatkhonova, M. (2025). THE ROLE OF INNOVATIONS AND INVESTMENTS IN THE SUSTAINABLE DEVELOPMENT OF ENTREPRENEURS IN UZBEKISTAN. Modern Scientific Research International Scientific Journal, 3(2), 94-97. https://doi.org/10.5281/zenodo.16157291
Jahon Banki. O‘zbekiston iqtisodiy ko‘rsatkichlari, 2023 yil.
McKinsey & Company. (2022). Innovations in Sustainable Business Practices
OECD. (2023). Small and Medium Enterprises and Innovation.
O‘zbekiston Respublikasi Innovatsion rivojlanish strategiyasi (2022-2030)
Schumpeter, J. (1942). Capitalism, Socialism and Democracy.
UNCTAD. (2022). Investment Trends and Policy Frameworks.
World Economic Forum. (2023). The Future of Sustainable Business Models.
Paul, J. (2015). Does international business education at the pre-experience level make a difference? Journal of International Business Studies, 46(9), 984–1003. https://doi.org/10.1057/jibs.2015.35
Acs, Z. J., & Preston, L. (1997). Small and medium-sized enterprises, technology, and globalization: Introduction to a special issue on small and medium-sized enterprises in the global economy. Small Business Economics, 9(1), 1–6. https://doi.org/10.1023/A:1007927602067
Ahmad, M., Rauf, A., Zeb, A., & Malik, M. S. (2022). Globalization and entrepreneurship: An empirical investigation in developing countries. PLOS ONE, 17(3), e0265757. https://doi.org/10.1371/journal.pone.0265757
The sun was shining brightly, it was the middle of summer…
I woke up early and was sweeping the yard. My grandmother was sitting and watching me.
“My girl, you studied all night, you didn’t sleep at all. You should rest a little!” she said with concern.
“It’s okay, Grandma, if I go one night without sleep, nothing will happen. God willing, I’ll pass the entrance exams, and then I’ll have plenty of time to rest.”
“If your parents were alive, they would have helped you,” she said sadly.
Just then, my stepmother woke up and came out.
“Ugh… what an unpleasant morning. Hey, Malika, did you iron your brothers’ shirts?”
“Not yet… I haven’t finished.”
“What a useless girl your mother gave birth to! When will you ever do things properly? You can’t do anything right—only know how to eat and sleep,” she shouted.
“Why do you keep scolding her, daughter-in-law? Leave her be. She’s already struggling so much,” Grandma defended me.
“Don’t interfere! It’s all your fault—spoiling her like this!” my stepmother retorted.
—
That’s how my days passed—my stepmother never stopped scolding me. My grandmother, however, helped me in every way she could and wished for me to be an educated, learned girl. I never gave up; I did my best.
I believed that one day I would become a great person and free both myself and my grandmother from this darkness. I lived with that determination.
If I managed to finish the housework one day, I wouldn’t finish my lessons; if I finished my lessons, the housework would remain. My stepmother’s temper worsened day by day. Not only me—she would even scold and quarrel with my grandmother. Every day, she found an excuse to say:
“When will I get rid of this witch of an old woman? She hasn’t died yet, and I haven’t been freed from her!”
Since my stepmother was the only one working in the house, our household wasn’t in good shape. She spent most of her earnings on herself and her two sons. To buy books and notebooks for my studies and medicine for my grandmother, I sometimes went to the evening market to sell small goods. Somehow, I managed.
One morning, while making breakfast, I was putting the kettle on when my stepmother said:
“Hey, Malika. Why are you so slow? Here—this is my most expensive dress. Iron it carefully, understand?”
Then she looked at me mockingly:
“Why are you staring like that? Right, you’ve probably never even seen clothes like these.”
I stayed silent and started ironing. While I was ironing, my grandmother said:
“Malika, dear, your tea must be boiling—go check on it.”
In my hurry, I left the iron on the dress without realizing it, and the dress got burnt.
What followed was chaos and shouting that’s hard to describe. My stepmother was so furious that she practically lifted the whole house with her screaming. I was standing in the kitchen trembling so hard that I didn’t even notice I had spilled boiling water on my own hand.
Still yelling, she came at me with fury in her eyes. She had always looked for a reason to hurt me, and now she had found one. She dragged me into the room where I had been ironing. Without thinking twice, with her eyes red with rage, she pressed the hot iron onto my right hand.
“That’s for defying me and burning my dress on purpose!” she shouted.
The pain was unbearable—I screamed and cried, writhing in agony.
—
Well, those days passed, but…
The questions Why? For what? What was my fault? still haunted me.
Years went by—it’s been two or three years since that incident. I entered university. God willing, I wanted to become a lawyer. Now I’m in my third year, living in the dormitory in Tashkent, far from my grandmother’s love. I miss her terribly. I call her occasionally from my friend’s phone, but I still miss her deeply—I haven’t seen her for almost eight months.
When we talk, she always says:
“I’m fine, my dear. I take my medicine on time. Don’t worry about me. Focus on your studies. You must become the lawyer I always dreamed of. Don’t think about me.”
But I knew my stepmother was still making her life miserable. I constantly worried—was she healthy? Was she eating properly? Was the house peaceful? These thoughts disturbed me even in class.
What could I do? If I went home, I’d fall behind in my studies. Oh God, please give me patience…
—
Finally, I finished my third year. Now I was in a taxi on my way home—so happy, so excited to see my grandmother again and feel her love. Not even the difficulties of the long journey could spoil my mood.
After two days of travel, I finally arrived at my street. I ran from the corner in excitement—but for some reason, the front of our house was full of people. The men wore traditional coats and skullcaps.
I was puzzled—what was happening here?
Then I heard someone say, “She has passed away, my girl.”
Those words were enough to make me lose myself. Everything blurred before my eyes, and the only thing I could hear was that sentence repeating in my head. I couldn’t say a word—I just sank to my knees at the gate.
When people tried to bring me back to my senses, I suddenly ran inside…
This cruel world had not only taken my parents from me—it had taken my grandmother too. I could only imagine how much pain she endured without me…
—
Years passed. I fulfilled my grandmother’s wishes and my own dreams. Perhaps they were her last will—of that, I have no doubt.
Now, I work as a judge in the Tashkent city court. I believe there are two main reasons why a girl would choose such a profession:
«First, repeated encounters with injustice in her life.
Second, a strong sense of pride in her homeland»
I had both reasons. And besides, this was my greatest goal—my duty to my grandmother and my country.
Today, as I sit by my grandmother’s grave, I recall the events of the past, one by one.
“Grandma… I’m here! Your beloved granddaughter has come to you as a judge. Forgive me for leaving you alone. I couldn’t save you from that place. But now, I will not allow such things to happen again. No child’s life will be like mine. I promise—and I will do everything to keep that promise.”
Homeland… This word is precious and sacred for every person. It embodies the land where we were born and raised, the memory of our ancestors, Uzbek traditions, language, and culture. The Homeland is our past, our present, and our future. Its praise is a feeling eternally sealed in every heart.
The praise of the Homeland is not only words spoken with the tongue but emotions flowing from the heart and manifested in our every action. Praising the Homeland means loyalty to one’s country, dedication to its prosperity, preserving national values, and living with a sense of Uzbek pride and honor.
Uzbekistan is our beloved Homeland. It is famous worldwide for its beautiful nature, rich history, unique culture, hardworking and hospitable people. Our ancient cities – Samarkand, Bukhara, Khiva, Shahrisabz – have for centuries been centers of science, culture, and craftsmanship. These cities have produced great scholars, thinkers, poets, and scientists whose priceless legacy remains a source of pride for us today.
During the years of independence, Uzbekistan has achieved great successes. Fundamental changes have been made in political, economic, social, and spiritual spheres. Our national values have been restored, religious freedom ensured, and our language has been granted the status of the state language. Significant achievements have been made in education, science, culture, and sports. Uzbekistan has found its place in the international arena and established friendly relations with many countries around the world.
Today, Uzbekistan is stepping into a new stage of development. Large-scale reforms are being implemented, our economy is growing, and the standard of living of our people is rising. We face great tasks ahead: to make Uzbekistan one of the most developed, prosperous, and flourishing countries in the world, and to leave a worthy legacy for future generations.
To accomplish these tasks, each of us must contribute. We must master our professions, work honestly, seek knowledge, develop ourselves, remain loyal to our country, preserve our national values, and live with a sense of Uzbek pride and honor.
The praise of the Homeland is the foundation of our unity, harmony, peace, and solidarity. We must be proud of being the children of one land, support one another, and work together for the prosperity of our country. Only then will we achieve our goals and turn Uzbekistan into one of the most developed nations in the world.
In conclusion, praising the Homeland is a feeling eternally sealed in every heart. Preserving it and showing it through our actions is our sacred duty. Uzbekistan is our beloved Homeland — may its praise forever echo in our hearts!
Ozodbek Narzullayev Student of Grade 11, School No. 45, Koson District, Kashkadarya Region
A second-year student in the Astronomy program at Termiz State University
Annotation:
This article explores the internal structure of Saturn, the second-largest planet in the Solar System. It examines the composition and characteristics of its core, metallic hydrogen layer, molecular hydrogen envelope, and atmospheric features. Special attention is given to the processes influencing Saturn’s internal heat, magnetic field, and rotation. The study also discusses current scientific models and data obtained from space missions, which contribute to a deeper understanding of the planet’s formation and evolution
Kalid so’z:
Keywords: Saturn, internal structure, core composition, metallic hydrogen, planetary science, gas giants, atmosphere, magnetic field.
The Internal Structure of Saturn
Saturn is the second-largest planet in the Solar System, with a diameter of approximately 120,536 km and a mass about 95 times that of Earth. It belongs to the class of gas giants.
1. Core
Saturn is believed to have a solid core at its center.
Estimated diameter: around 25,000 km.
Composition: iron, nickel, oxygen, silicates, and ice mixtures.
Estimated temperature: ~11,700 °C.
2. Metallic Hydrogen Layer
Located above the core.
Under extreme pressure and temperature, hydrogen takes on a metallic form. The planet’s strong magnetic field is generated by movements within this layer.
3. Molecular Hydrogen Layer
Composed mainly of molecular hydrogen mixed with helium.
Temperature increases with depth.
4. Atmosphere
Main composition: 96% hydrogen, 3% helium, with traces of methane, ammonia, and other gases.
Cloud layers are divided into three main zones from top to bottom:
Ammonia ice clouds Ammonium hydrosulfide clouds
Water ice clouds
5. Heat Source
Saturn emits about 2.5 times more heat than it receives from the Sun.
This is due to the Kelvin–Helmholtz process (gravitational compression) and helium rain.
6. Research
Voyager 1 and Voyager 2 (1979–1981) provided the first comprehensive data about Saturn.
Cassini–Huygens (2004–2017) gathered highly detailed information about Saturn, its rings, and moons.
Saturn or Zuhal (Arabic) — one of the nine major planets of the Solar System; the sixth planet from the Sun in terms of distance. Its astronomical symbol is ♄. The average distance from the Sun is 9.58 astronomical units (1,429.3 million km). In terms of size, it ranks second after Jupiter. Due to its orbital eccentricity of 0.056, this distance decreases at perihelion and increases at aphelion. Saturn completes one full revolution around the Sun in 29 years and 167 days. Its rotation period at the equator is 10 hours 14.5 minutes, increasing toward the poles, as on Jupiter. Saturn’s average radius is 58,000 km, and at the equator it is close to 60,000 km. Its density is 0.72 g/cm³ (about 0.13 of Earth’s density), and its mass is 5.681 × 10²⁷ g (95.28 times that of Earth). It is composed mainly of hydrogen and helium. The inclination of Saturn’s equator to its orbital plane is 26°45′. Elongated dark belts along the equator and white spots can be observed on its surface.
Spectroscopic studies have shown that Saturn’s atmosphere is mainly composed of hydrogen (H₂) and methane (CH₄); later, acetylene (C₂H₂) and ethane (C₂H₆) were also discovered. Infrared observations indicate a temperature of about 95 K. Saturn emits regular radio waves and has its own magnetic field. Even at such low temperatures, methane and ammonia remain in a gaseous state in its atmosphere. Saturn was first observed by Galileo in 1610.
Saturn’s most remarkable feature is its ring system, which lies in the plane of its equator. The rings are mainly composed of three distinct concentric parts: the innermost S ring, the middle V ring, and the outer A ring. All rings are almost transparent, allowing the stars behind them to be seen. The gap between the A and V rings is called the Cassini Division. The rings are a few kilometers thick, have a diameter of about 250,000 km, and are roughly 100 million years old. As early as the beginning of the 20th century, it was established that the rings are not solid, but consist of numerous solid particles orbiting Saturn according to Kepler’s laws. Their infrared spectra are very similar to those of ice and frost.
Saturn has 22 known natural satellites. The largest among them are Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, Iapetus, Phoebe, and Janus. Titan, with a diameter of 4,850 km, is one of the largest satellites in the Solar System (larger than the Moon). Titan possesses an atmosphere composed of methane and hydrogen. Of Saturn’s moons, only Phoebe moves in a retrograde orbit relative to the others.
Saturn is the sixth planet from the Sun and the second largest in the Solar System. Along with Jupiter, Uranus, and Neptune, it belongs to the class of gas giants. It is named after the ancient Roman god Saturnus (Unicode: ♄).
Saturn is composed mainly of hydrogen, with small amounts of helium and other elements. The region near its core consists of rock and ice, surrounded by metallic hydrogen. Its outer atmosphere is generally uniform in appearance, although persistent storms sometimes occur. Winds on Saturn can reach speeds of up to 1,800 km/h, which is much faster than those on Jupiter. Saturn has a magnetosphere weaker than Earth’s but stronger than Jupiter’s.
Saturn is famous for its rings, which are made primarily of ice particles mixed with dust. Many moons orbit the planet — about sixty are currently known. Titan, Saturn’s largest moon and the second largest in the Solar System (after Ganymede), is larger than Mercury and is the only natural satellite in the Solar System known to have a dense atmosphere.
Foydalanilgan adabiyotlar:
1. Mamadazimov, M., and A. B. Narbayev. “An e-learning guide for students of the 11th grade of secondary education and secondary special, vocational education institutions.” (2018).
2. Bahramovich N. A. STRUCTURE AND DIDACTICAL POSSIBILITIES OF THE ELECTRONIC TRAINING MANUAL ON ASTRONOMY DEVELOPED FOR PUPILS OF THE 11TH GRADES OF SECONDARY SCHOOLS BASED ON MEDIA EDUCATION //Archive of Conferences. – 2020. – Т. 5. – №. 1. – С. 40-42.
3. Narbaev A. B. Advantages of using media technology in astronomy teaching //Journal of Innovations in Pedagogy and Psychology. – №. 2. 4. Narbaev A. B. METHODS OF USING MEDIA EDUCATION AND TELECOMMUNICATION TECHNOLOGIES IN TEACHING THE TOPIC” VISUAL MOTION OF THE SUN AND STARS” //Central Asian Problems of Modern Science and Education. – 2020. – Т. 2020. – №. 1. – С. 119-125.
4. Ksanfomaliti L. V., Planeti otkritie zanovo, M., 1978; Kulikovskiy P. G., Spravochnik lyubitelya astronomii, M., 1971; Silkin B.I., V mire mnojestva lun, M., 1982; Mamatazimov M., Quyosh sistemasi haqida ocherklar, T., 1984
