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Wednesday, July 23, 2014

Systems of Linear Equations



Linear Equations in n variables x

 \begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& b_m \\ \end{alignat}

x_1,\ x_2,...,x_n are the unknowns
 a_{11},\ a_{12},...,\ a_{mn} are the coefficients of the system
 b_1,\ b_2,...,b_m are the constant terms

Linear equations have no products or roots of variables and no variables involved in trigonometric, exponential, or logarithmic functions. Variables may appear only to the first power.

Examples:
        Linear Equations                                                 Nonlinear Equations

  • x + 3y = -4\                                    xy + 7 = 2\
  • 7x_1 = 15 + x_2\                                  \sqrt{x_1} + x_2 = 11
  • z\sqrt{2} + e = \pi\                                     x^3 = 6 - 12z\
A solution of a linear equation in n  variables is a sequence of n real numbers (s_1,s_2,....,s_n)\  which satisfies the linear equation. 
Example:
                                     \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}   
This has a solution of (1,-2,-2)\

The set of all solutions of a linear equation is called its solution set, and when the this set is found, the equation is said to have been solved. To describe the entire solution set of a linear equation, a parametric representation is often used, as shown below.

 Example 1. Solve the linear equation x + 2y = 4
Solution     To find the solution set of an equation involving two variables, solve for one of the variables in terms of the other variable. If you solve for x in terms of y, you obtain
                     x = 4 - 2y
In this form, the variable y is a free variable, which means that it can take on any real value. The variable x is not free because its value depends on the value assigned to y. To represent the infinite number of solutions of this equation, it is convenient to introduce a third variable t called a parameter. By letting y = t, you can represent the solution set as
               x = 4 - 2t,           y = t, t is a real number.
Particular solutions can be obtained by assigning values to the parameter t. For instance, t = 1 yields the solution x = 2 and y = 1, and t = 4 yields the solution x = -4 and y = 4.


Example 2. Solve the linear equation 3x + 2y - z = 3
Solution     Choosing y and z  to be the free variables, begin by solving for x to obtain
                         3x = 3 - 2y + z
                          x = 1 - 2/3y +1/3z,
Letting y = s and z = t, you obtain the parametric representation
                      x = 1 - 2/3s + 1/3z,             y = s,          z = t
           where s and t are any real numbers


For any linear system of equations there are three possibilities regarding solutions:
1. Consistent System

  • Consistent Dependent System - the system has exactly on solution.
  • Consistent Independent System - the system has an infinite number of solutions.
2. Inconsistent System - the system has no solution.


Posted by Unknown at 6:03 AM

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