# Full Project – Design and implementation of an automated four-in-one algebra solver system

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**CHAPTER ONE**

**1.1 INTRODUCTION**

Money spent on computers in public schools has increased at a steady rate over the last 20 years. Even with this increase in expenditure on computers, there is remarkable potential for the effective integration of computers in schools mathematics, focusing on what can be done above and beyond with pencil and paper alone (pea 1986). Using computers as cognitive tools to assist student in learning powerful mathematics that they could have approached without the technology should be a key goal for research and development – not only learning the same mathematics better, stronger, faster but also learning fundamentally different mathematics in the process. Different Educational Software Program exist providing students and teachers with option of not only learning from their books but also learning from this technology. This software incorporates four different mathematics topics into one automated system allowing mathematical problems in these four categories to be solved.

**1.2 BACKROUND OF STUDY**

**Linear Algebra**

The study of linear algebra includes the topics of vector algebra, matrix algebra and the theory of vector spaces. Linear algebra originated as the study of linear equations including the solution of simultaneous linear equations, an equation is linear if no variable in it is multiplied by itself or any other variable. Thus the equation (3x + 2y + z = 0) is a linear equation in three variables. The equation (x^3 + 6y + z + 5 = 0) is not linear because the variable x is raised to the power 3 (multiplied together three times); it is a cubic equation. The equation ( 5x – xy + 6z = 7) is not a linear equation either because the product of two variables (xy) appears in it. Thus linear equations are always degree 1.

Two important concepts emerge in Linear Algebra to help facilitate the expression and solution of systems of Simultaneous Linear Equations. They are the Vector and Matrix. Vectors correspond to directed line segments. They have both magnitude (length) and direction. Matrices are rectangular arrays of number. They are used in dealing with the coefficient of Simultaneous Equations using Vector and Matrix notation, a system of Linear Equation can be written in the form of a single equation, as a Matrix times a Vector.

Linear Algebra has a wide variety of applications. It is useful in solving network problems, such as calculating current flow in various branches of complicated electronic circuits, or analyzing traffic flow patterns on city streets and interstate highways. Linear algebra is also the basis for a process called Linear Programming widely used in business to solve a variety of problems that often contain a very large number of variables.

Since the solution to a system of Simultaneous Equations, as pointed out earlier, correspond to a point in space where their graph intersect in a single point and since vectors represent points in space, the solution to a set of Simultaneous Equations is a Vector. Thus all the variables in a system of equations can be represented by a single variable, namely Vector.

**QUADRATIC EQUATION**

A Quadratic Equation is a second order polynomial equation with a single variable (x)

Ax^2 + bx + c = 0 where a ≠ 0

On clay tablets dated between 1800 BC and 1600 BC, the ancient Babylonian left the earliest evidence of the discovery of quadratic equations and also gave early methods for solving them. Indian mathematician Baudhayana who wrote a sulba sutra in acient India circa 8^{th} century BC first used quadratic equations of the form

ax^2 = c and ax^2 + bx = c

and also gave methods for solving them.

Babylonian mathematician from circa 400 BC and Chinese mathematician from circa 200 BC used the method of completing the square to solve quadratic equation with positive roots, but did not have a general formula, Euclid, a Greek mathematician, produced a more abstract geometric method around 300 BC. The first mathematician to have found negative solutions with the general algebraic formula was Brahmagupta (India 7^{th} century). Mohammed ibn musa al kwarizmi (Persia, 19^{th} century) developed a set of formula that worked for positive solutions. Bhaskara II (1114-1185), an Indian mathematician astronomer, solved quadratic equations with more than one unknown and is considered the originator of the equation.

**SERIES**

A series is a set of numbers such as: 1 + 2 + 3 which has a sum. A series is sometimes called a progression like “arithmetic progression”.

**SEQUENCE**

A sequence, on the hand is a set of numbers such as: 2,1,3 where the order of the numbers is important. A different sequence from the above is: 1,2,3

A series such as 1+2+3… has the same sum as 2+1+3 but the numbers are in a different sequence

**ARITHMETIC SERIES**

A pure Arithmetic Series is one where the difference between successive terms is a constant. We can call the constant “d”. if the first term is “a”, then the Arithmetic Series is: a + (a+d)+(a+2d)+…(a+(n-1)d)

**GEOMETRIC SERIES**

A pure Geometric Series or Geometric Progression is one where the ratio “r” between successive terms is a constant. Each term of a Geometric Series, therefore involves a higher power than the previous term.

Algebraically, we can represent the n terms of the geometric series with the first term a, as:

Sn= a+ar+ar^2+ar^3+…ar^n-1

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**COMMERCIAL ARITHMETIC**

Simple Interest: Simple Interest is money you can earn by initially investing some money (the principal). A percentage (the interest) of the principal is added to the principal making your initial investment grow

Simple Interest is given by

I = (P * R * T)/100

P = principal, R = rate, T = number of years or duration of time

Compound Interest: This is interest added to the principal of a deposit or loan so that the added interest also earn interest from then on, the addition of interest to the principal is called Compounding, Compound Interest is given by

A = P(1 + r/n)^nt

P = principal

r = rate

t = number of years or duration of time

n = number of time the interest is compounded per year

A = amount accumulated after n years, including interest

**1.3 AIM OF STUDY**

The aim of this project is to:

- Design and implementation an application software using visual basic 12 to solve problems involving
- Simultaneous linear equation
- Quadratic equation
- Sequence and Series and
- Commercial arithmetic.

**1.4 OBJECTIVES OF STUDY**

The major objective for designing this system includes:

- To Design a program that combines four selected algebra topics in Secondary School Mathematics into one solvable system.
- Design an automated system that answers can be verified from even after solving the questions manually.
- To provide schools with software that assist in teaching and learning of mathematics
- To broaden student knowledge of computer usage and it application
- To provide a software program that makes solving of mathematical problem easy, fast and interesting.
- Design a system that is dependable by eliminating error in calculation to the beeriest minimum and provide high productivity and enhancement in terms of processing time and result delivery

**1.5 SIGNIFICANCE OF STUDY**

The design and implementation of this system will go in a long way to enhance Teaching and learning in schools as it will

- Make it fast and efficient in obtaining solutions to questions in this four categories
- Considering this system is a fusion of four different categories of mathematics topics into one solvable system, it use and significance cannot be overemphasized.
- It will broaden student knowledge of computer system and how it has been use to provide answers to problems in mathematics and other fields
- This system provide a framework to help teachers understand the ways in which student can learn from or with software and provide them opportunities to support different types of student learning.

**1.6 STATEMENT OF PROBLEMS**

- The need to implement system that can give accurate result at a fast rate
- The need to have teaching aid that will assist in teaching and learning in our schools
- The need to make problem easier to solve.
- The need to have a systems where answers to mathematical problems can be verified from even after solving them manually
- The need to support different types of student learning.

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**1.7 SCOPE OF STUDY**

The design of this system only covers the implementation of a program that can be use to find solution to problems in the following Mathematics Algebra topics:

- Simultaneous linear equation
- Quadratic equation
- Sequence and series (arithmetic & geometric progression) and
- Commercial arithmetic (simple and compound interest).

**1.8 LIMITATION OF STUDY**

This research work is limited to the implementation of software for the solution of mathematical problems involving four categories which includes: simultaneous linear equation, quadratic equation, arithmetic progression, geometric progression, simple interest and compound interest furthermore the possibility of integrating additional mathematic topics into the system do exist and the system is applicable only when the desirable parameters for solving are given.

**1.9 DIFINITION OF TERMS**

- Linear equation: This is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
- Quadratic equation: A Quadratic Equation is a second order polynomial equation with a single variable (x)
- Simple interest: Simple Interest is money you can earn by initially investing some money (the principal). A percentage (the interest) of the principal is added to the principal making your initial investment grow
- Compound interest: This is interest added to the principal of a deposit or loan so that the added interest also earn interest from then on
- Constant: In algebra a constant is a number on it own, or sometimes a letter such as a,b or c to stand for a fixed number.
- Variable: A variable is a quantity that can change or that may take on different values.
- Roots of an Equation: this are values of the variable, that turn equation into correct equality
- Slope: the slope of a line is the difference between one point (x1,y1) and the next point on the line, (x2,y2). The difference is represented as (y2-y1)/(x2-x1)
- Intercept: this is the point or region where two or more line(representing an equation) meet
- Progression: This is a sequence of number where each subsequent term or number is well defined in a particular order
- First term: this is the first number or value in a progression

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## Full Project – Design and implementation of an automated four-in-one algebra solver system