Chapter 1 - Chapter 1 Linear Equations vs. Nonlinear...

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Chapter 1 Linear Equations vs. Nonlinear Equations Definition In general, a linear equation with n unknowns is an equation that can be placed in the form a 1 x 1 + a 2 x 2 + a n x n … = b In the equation above, a have known values and are called coefficients ; b also has a known value and is called the constant for the equation, while x are the variables or unknowns . The key characteristic of a linear equation is that the variables appear to the first power only. In particular, a linear equation never contains products of variables (such as x 1 x 2 ), functions of variables (such as sinx 2 ), or powers of variables (such as x 3 5 ). Consistent Systems and Inconsistent Systems We will frequently refer to a system of m linear equations in n unknowns as an (m x n) linear system. If m = n (so that there are exactly as many equations as there are unknowns) we will call the system a square system. If a system is not square , it is called a rectangular system. Definition If an (m x n) system has at least one solution, then we say the system is consistent . If the system has no solutions whatsoever, then it is called inconsistent . Elementary Operations in 2 x 2 Systems This module introduces Gauss-Jordan elimination , a practical and systematic method for solving systems of linear equations. We will restrict ourselves in this module to square systems that have unique solutions. Later, in Module 1.2 and 1.3, we will treat rectangular systems and systems that have no solutions or infinitely many solutions. The solution procedure we want to discuss, Gauss-Jordan elimination, is based on an idea that is probably familiar to you--that of using elementary operations to eliminate some of the variables from an equation. Elementary Operations 1. Interchange any two equations ( ) 2. Multiply both sides of any equation by a nonzero constant ( )
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3. Replace any equation by adding a constant multiple of any other equation to it ( ) It can be shown that if we use any of the three elementary operations to create a new system, then the new system has exactly the same solutions as the original system. We call two systems having the same solution set equivalent systems . Thus, whenever we apply an elementary operation, we produce a system equivalent to the one we started with. The Solution Possibilities for a 2 x 2 system Geometric representations of (2 x 2) linear systems The graph of a linear equation in two unknowns is a line in the plane. Thus, a system of two linear equations in two unknowns can be represented geometrically by two lines in the plane. The solution to the system is found at the intersection of the two lines.
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Chapter 1 - Chapter 1 Linear Equations vs. Nonlinear...

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