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Patterns Withing Systems of Linear Equations

The usual letter for the unknown number is. A real problem can be written as: This is called an equation because there is a sign. In order to find the value of the unknown number, algebra’s rules can do whatever it likes to this equation as long as it does the same to both sides of the equation. So far it has had equation with a single unknown number. What if it has two unknown numbers? In fact, an equation with two unknown has an infinite numbers of pairs of answer. To fix a single pair of number as the answer, it needs another equation. A pair of equation, each with two unknown numbers is called simultaneous equations. They can be solved together to give the values for the unknowns that satisfy both equations simultaneously. This paper contains a mathematical research about systems of linear equation when their coefficients obey arithmetic or geometric progressions. An arithmetic progression is a sequence of numbers where each number is a certain among larger than the previous one. The numbers in the sequence are said to increase by a common difference, d. For example: is an arithmetic progression where the. The term of this sequence is. On the other hand, a geometric progression is a sequence where each number is times larger than the previous one. is known as the common ratio of the progression. The term of a geometric progression, where is the first term and is the common ratio, is: . For example, in the following geometric progression, the first term is , and the common ratio is : the term is therefore. The purpose of this portfolio is to show how with the aid of technology using appropriate computer software likes Autograph and Maxima packages (see Figure 1) is quick and easy to get graphical representations of algebraic equations. Thus, how in many situations, the graphs offers much more insight into the problem than does the algebra. Part A will consider the patterns within systems of linear equations:, where and are in arithmetic progression. While, in Part B the same coefficients obey geometric progression. Part A. System of linear equations formed with arithmetic progressions. Arithmetic progressions In algebra, letters are used in place of numbers that are not known. The usual letter for the unknown numbers are or . . The numbers are constants in an equation, for example: For instance in the above equation, and are known as constants in the equation. It says that the constant and form a arithmetic progression if they have a common difference, such as: Constants in a system of linear equations Given the system of linear equations. The coefficients are detected as follow: Examining the first equation, it sees a pattern in the constants of the equation. i.e. is the constant preceding the variable , and precede and the equation equals 3. The constant have a common dif
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