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.

                                     References:

·  Azlarov, T., & Mansurov, Kh. (1994). Mathematical Analysis, Vol. 1. Tashkent: “O‘qituvchi”. 315 pages.

·  Azlarov, T., & Mansurov, Kh. (1995). Mathematical Analysis, Vol. 2. Tashkent: “O‘qituvchi”. 336 pages.

·  Ayupov, Sh. A., & Berdikulov, M. A. (2004). Theory of Functions. Tashkent: “O‘AJBNT” Center. 148 pages.

·  Turgunbayev, R. (2008). Mathematical Analysis, Part 2. Tashkent: TDPU.

·  Jo‘rayev, T., et al. (1999). Fundamentals of Higher Mathematics, 2nd ed. Tashkent: “O‘zbekiston”.

Atlantis

Grant her the trident 

with which to conjure up the sunken city,

the square and the compass

to calculate the diameters of the cerebral journey,

as her ark is bent on pursuing

the emerald of a charted symmetry.

Grant her the trident

with which to subdue the dragon

that had been long conceived

in the depth of her contaminated heritage,

as her crusade is bent on surmounting

the convolutions of a mental labyrinth.

Into the Abyss

It will take the seven oceans to cleanse the soiling of our souls,

to flush out the debris from our clogged pores,

to peel off the ugliness

that drapes our tarnished walls,

the soot, the mould.

There are no Charles Darneys in the real world,

a noble spirit that would sacrifice its life

to save a scapegoat’s,

that is plunging down into the abyss

once and for all.

A Visitation

In my world, there are no kings and queens,

hence the concept of monarchy is alien to me,

and this lack of interest

is not intended to manifest

any disrespect

for the royal sect.

In a dream, I descend a flight of ancient steps,

only to view a partly dilapidated wing

of a majestic building,

where I am told by a dark-skinned Usherer

I once had my own dwelling.

At the huge doorway, a young woman,

who wears a white, woolen hat

and a very beautiful shawl,

embraces me with tears of joy.  

The blueness of her eyes vies

with the azure of the skies.

In the morning, I start to wonder at the capacity of our dreams

to evoke people who have no presence in our reality,

but a year later a picture of the woman in her youth

appears on my timeline on Facebook.

I still ponder over what makes a monarch bid me goodbye

three days before she dies?

An Encounter II

I carry my dog five flights of stairs

four times a day,

and as I breathlessly mount the arduous steps

I say to Lucia “the sniper has not caught up with us yet,”

then I plant three kisses on her tiny, velvety head.

But don’t snipers prefer to maintain some distance

between themselves and their intended victims?

I resolve to ascertain this fact on the net

since this topic is still alien to my literary mindset!

Fragrance

Let me remind you that it’s the head that teems with scents,

not thy nostrils!

They only titillate its mucous for fleeting seconds,

or some lingering minutes,

but have a lasting impact upon your cerebral cells

for as long as you live.

Each scent has its own personal context

an emotional aura,

conjuring up the past

and whatever pertains to thy daily presence,  

a fragrant image

that brings to life all that is aesthetic

and hauntingly pleasant.

a planetary wrong turn

with no off-ramp to the Goldilocks zone

       *

the one-eyed dogcatcher told the boy about a little green farm

       *

he prefers the original artificial taste of Rocket Pops

       *

when translating ‘summer thunder’ was a cinch

       *

locust swarms on the I-80 corridor

the HOV lane transcends

       *

the urge to answer the unknown caller

       *

before Les Mots

I played Wipe Out

on the surface of the sun

       *

a seance of pewter rain falling

an inch at a time

       *

I rushed in to tell about the silver maple leaf

independently astir in the heat 

       *

pennies on the track

how I got to know the long faces of Lincoln

       *

he couldn’t hear the bell crickets

with the air conditioner on full blast

       *

breathing the same air as the bulked-up horsefly

The Footprints of Borges

In labyrinths of ink and knowledge,

a blind guide, illuminating being.

With words, he wove parallel worlds,

mirrors where we see our longings.

In every library, a universe,

on every page, a perverse dream.

Time is an eternally branching river,

and destiny, a garden without winter.

His footprints, shadows in the Aleph,

an echo in the garden of intersecting paths.

In every verse, an echo of his faith,

Borges lives, where reality blurs.

GRACIELA NOEMI VILLAVERDE is a writer and poet from Concepción del Uruguay (Entre Rios) Argentina, based in Buenos Aires She graduated in letters and is the author of seven books of poetry, awarded several times worldwide. She works as the World Manager of Educational and Social Projects of the Hispanic World Union of Writers and is the UHE World Honorary President of the same institution Activa de la Sade, Argentine Society of Writers. She is the Commissioner of Honor in the executive cabinet IN THE EDUCATIONAL AND SOCIAL RELATIONS DIVISION, of the UNACCC SOUTH AMERICA ARGENTINA CHAPTER.

Annotation

This article discusses the attention and opportunities directed by our state towards women, young people with disabilities, and socially vulnerable groups of the population so that the youth of our society can grow up as healthy, educated, hardworking, and independent-minded individuals.

Key words: Society, youth, neighborhood, state, gender equality, power, women, leader.

Today, young people have become a significant part of society who form state authority, influence the internal and external policies of the country, and actively participate in its activities. The role of youth in the relations between the state and society is noticeably increasing. In Uzbekistan as well, this indicator is steadily growing.

The President of the Republic of Uzbekistan, Shavkat Mirziyoyev, emphasized:
“We will continue the state policy regarding youth firmly and consistently. Moreover, we will not only continue but also raise this policy to the highest level as one of our most important priorities, in line with modern requirements. For our youth to become independent thinkers, with high intellectual and spiritual potential, who are not inferior to their peers in any field worldwide, and to live happily — our state and society will mobilize all their strength and opportunities.”
[Mirziyoyev Sh.M. Together We Will Build a Free, Prosperous and Democratic Uzbekistan. Tashkent: “Uzbekistan,” 2016, p.14].

This is one of the greatest examples of the attention given to youth.

Since the future, development, and progress of countries are in the hands of young people, it is important to note the international concepts and terms regarding youth. Each country sets its own age limits for youth. For example, in the USA and Japan, people aged 13–14 up to 29–30 are considered youth; in Bulgaria, 15 to 29; in Russia, 15 to 30; in Germany, “younger youth” are 14–18 and “older youth” are 18–27. According to Article 3 of the Law of the Republic of Uzbekistan “On State Youth Policy,” citizens from the age of 14 up to 30 are classified as youth [http://blog.xabar.uz/post/yoshlar-orasida-huquqiy-savodxonlikni-oshirishda-huquqiy-ta-lim].

This shows that a large portion of the population of Uzbekistan consists of youth. It is therefore necessary to focus on the opportunities provided to them.

In Uzbekistan, various opportunities are being created to enhance the role of youth and raise them as important figures for the future of the state. For instance, democratic reforms were introduced into the management system of neighborhood institutions. New management mechanisms were created, a vertical system for working with youth was established, and problems of young people are now addressed directly within neighborhoods. In order to increase the efficiency of educational and upbringing activities, the Decree of the President of the Republic of Uzbekistan “On Measures for the Fundamental Improvement of the System of Working with Youth in Neighborhoods” (PQ-92, January 19, 2022) was adopted [https://lex.uz/uz/docs/-5831865].

According to this decree, the position of “Youth Leader” was introduced in every neighborhood. Databases such as the “Youth Balance,” “Youth Register,” and the “Youth Portal” were established, with responsibilities assigned to youth leaders. Material assistance, housing subsidies, partial coverage of tuition fees, and support for vocational training and driver’s courses are also being provided.

A significant part of these opportunities is allocated to women. By the Presidential Decree “On Measures to Further Accelerate Systematic Support for Families and Women” (PF-87, March 7, 2022) [https://lex.uz/docs/-5899498], systematic reforms are being carried out to protect women’s rights and interests, increase their social, political, and economic activity, ensure gender equality, and engage women in entrepreneurship.

In the field of education, many opportunities are being created. For example, socially active youth receive privileges when applying to universities. A Presidential grant is awarded to the top 200 students with the highest entrance exam scores. Students studying under contracts are provided with education loans, and for active students or those who have completed military service, 35% of their tuition is reimbursed. The introduction of the “Youth Register” has also raised youth policy to a new stage [https://pm.gov.uz/oz/lists/view/2117].

As a result, thousands of unemployed youth in need of social, psychological, and economic support have been registered and provided with targeted assistance.

The First President of Uzbekistan, Islam Karimov, stated:
“The future of our Motherland, the destiny of our people, and the reputation of our country in the world community, first of all, depends on how our children will grow up, what kind of people they will become, and how they will enter life. We must never forget this truth.”
[Karimov I. High Spirituality is an Invincible Power. Tashkent: “Ma’naviyat,” 2008].

This means that it is essential to teach young people how to find their place in life and make effective use of the opportunities available to them.

Attention is also given to youth with disabilities. According to the Presidential Decree (PQ-74, February 27, 2023) [https://www.lex.uz/docs/6391378], individuals with disabilities and their caregivers are granted priority in receiving state services, as well as strengthened legal guarantees and legal support.

President Shavkat Mirziyoyev has noted:
“Building a New Uzbekistan is not just a desire or a subjective phenomenon, but an objective necessity based on historical foundations, current political, legal, socio-economic, and spiritual realities, and the age-old aspirations of our people. The driving force of our reforms is the knowledge, abilities, and potential of our citizens.”
[Mirziyoyev Sh. The Development Strategy of New Uzbekistan. Tashkent: “Uzbekistan,” 2022].

Conclusion

Indeed, the opportunities and attention provided to youth serve as the strongest motivator for us to move forward. If every young person knows how to properly use these opportunities, they will undoubtedly grow into specialists who contribute greatly to the prosperity of our country.

The expansion of educational institutions such as “Presidential Schools,” “Creative Schools,” and “Temurbek Schools” demonstrates the country’s commitment to raising competitive personnel. Youth are also being supported to start businesses through loans, provided with professional training, and ensured employment.

Great importance is also attached to sports, as our compatriots proudly raise the flag of Uzbekistan at international competitions. Special attention is paid to youth with disabilities, who also achieve remarkable success as athletes, artists, teachers, businessmen, and political leaders.

Thus, it is clear that opportunities in Uzbekistan are balanced for all citizens, regardless of whether they are healthy or disabled. This gives us, the youth, great pride and motivation.

The continuous reforms and conditions created for us inspire satisfaction and confidence in our future, ensuring that Uzbekistan develops into a prosperous state with educated, skilled, and active youth.

References

1. Mirziyoyev Sh.M. Together We Will Build a Free, Prosperous and Democratic Uzbekistan. Tashkent: “Uzbekistan,” 2016.
2. Karimov I. High Spirituality is an Invincible Power. Tashkent: “Ma’naviyat,” 2008.
3. Mirziyoyev Sh. The Development Strategy of New Uzbekistan. Tashkent: “Uzbekistan,” 2022.
4. http://blog.xabar.uz/post/yoshlar-orasida-huquqiy-savodxonlikni-oshirishda-huquqiy-ta-lim
5. Decree of the President of the Republic of Uzbekistan PQ-92, January 19, 2022. https://lex.uz/uz/docs/-5831865
6. On Improving the System of Working with Youth in Neighborhoods. http://lex.uz/uz/docs/-5831865
7. Presidential Decree PF-87, March 7, 2022. https://lex.uz/docs/-5899498
8. Youth of New Uzbekistan: Opportunities and Prospects. https://pm.gov.uz/oz/lists/view/2117
9. Decree of the President of the Republic of Uzbekistan PQ-74, February 27, 2023. https://www.lex.uz/docs/6391378

Childhood

Childhood — gone, never to return,

Yet it makes us long in innocence.

That pure, untainted childhood,

I still yearn for it even now.

Growing up is good, yes,

But childhood was the best.

Pain was only in the body,

There were no grudges at all.

Has it truly gone forever?

I’ll spend my life missing it.

Ah, it was so different,

Those wonderful days we miss.

We were children back then,

We knew nothing of sorrow or worry.

We played and were mischievous a little,

Yet still — we were children.

There were no thieves of the heart,

Pain was only in the body.

Everyone was good to us,

We knew nothing of the bad.

And as we grew older,

We learned everything.

If growing up is truly like this,

I’d have been content never to grow up!

Sharifova Saidaxon Kamolliddinjon qizi was born on May 26, 2008, in the village of Kaldushon, Furqat District, Fergana Region, into an educated family.

In 2015, she was admitted to the 1st grade of Secondary School No. 21 in Furqat District, where she is currently an 11th-grade student.

In addition to school textbooks, she attends various extracurricular clubs. Saidaxon is fluent in conversational English. Despite her young age, she is the holder of more than 15 international and official certificates and has actively participated in numerous projects. Her poems have been published by the official publishing house Lulu Press Inc.

THE ECONOMIC ROLE OF GLOBALIZATION IN THE RAPID DEVELOPMENT OF ENTREPRENEURS IN UZBEKISTAN

Madinabonu Mamatxonova Kamol qizi – 2^nd-year student, Faculty of Tourism and Economics, Department of Economics, Kokand University

mamatxonovamadinabonu60@gmail.com

     Abstract

     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

1. 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
2.  Jahon Banki. O‘zbekiston iqtisodiy ko‘rsatkichlari, 2023 yil.
3. McKinsey & Company. (2022). Innovations in Sustainable Business Practices
4.  OECD. (2023). Small and Medium Enterprises and Innovation.
5.  O‘zbekiston Respublikasi Innovatsion rivojlanish strategiyasi (2022-2030)
6.  Porter, M.E., & Kramer, M.R. (2011). Creating Shared Value. Harvard Business Review
7.  Schumpeter, J. (1942). Capitalism, Socialism and Democracy.
8.  UNCTAD. (2022). Investment Trends and Policy Frameworks.
9.  World Economic Forum. (2023). The Future of Sustainable Business Models.
10. 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
11. 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
12. 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

Electronic sources:

https://www.weforum.org/agenda/archive/sustainable-business-models

https://www.oecd.org/industry/smes-and-entrepreneurship/

https://mininnovation.uz/