Introduction:
A linear equation is an algebraic equation in which the highest power of the unknown variable is a unit. Linear equation is in the form of where a, b and c are constants and x and y are the variables of the equation (PBS. Mathline). A linear equation produces a straight line graph when plotted to scale on a graph paper.
Solving a System of Linear Equations
A system of equations is the case when we have more than one linear equation. (When there are two or more equations). For elementary studies, the number of unknowns in the system is equal to the number of equations in the system. We shall from here study a system of two (simultaneous) equations. There are a minimum of four known methods of solving a system of linear equations, namely; Substitution, Elimination (Addition), Matrix and Graphical Methods
We shall as suggested in the question look at the first three methods. We shall also illustrate each of the methods with a different example.
Substitution Method
Consider the system of two simultaneous equations
We take one convenient equation and select an easy variable which we use to express in term of the other terms. In this case we make x the subject in equation (2) and substitute in equation (1) i.e. . We substitute in equation (1). (1) Becomes. By simplifying we have. After getting the value of variable y, we substitute it again in either of the equations which convenient and easy to simplify in order to find the other variable x. using equation (2) . The pair of solutions is
Addition (Elimination) Method
We also illustrate this method of solving a system of two simultaneous equations using a different example. We choose a variable from a convenient equation to eliminate from the equations by adding (the magnitude of the coefficient of the variable should be the same in the two equations). In this example we can eliminate y by adding equations (1) to (2). We get. We now put the value of x in either of the equations we can solve easily. If we in (1), we have. The solution to the system is now
Matrix Method
Given a system of linear equations, we first transform the system in the matrix form (for coefficients, variables and constants). We determine the inverse of the coefficients which we pre-multiply to both sides of the equation. We now take an example. We consider the system. We write the matrices. We now determine of the coefficient matrix and find its inverse. Let. . The inverse we now pre-multiply the inverse through the equation and have
. And now the values of the system are. This satisfies the equations.
Inequalities
These are algebraic expressions in which one of the sides is greater than the other. There are symbols used in system which are less than (<), greater than (<), less than or equal to (atleast,) and greater than or equal to (at most, ≥). For example an expression and is a system of two linear equations.
Linear Programming
This is an explanation of inequalities by drawing graphs. This is applied to many real life situations. We shall use a system
We first find the points where the lines would pass i.e.
When. The line would pass through points (0, 1) and (2, 0).
For the line. When. When. The line would pass through (0, 2) and (0, 3). We now plot each of the lines on the graphs and shade the unwanted regions for the each of the four inequalities given above. After shading the unwanted regions, we expose the feasible (or accessible) region where the points whose values of x and y satisfy the system of linear inequalities or constraints or conditions of the system. This is called linear programming.
Conclusion
Linear programming deals with finding the optimum values (maximum or minimum) of linear expression of two or more variables which are related to a set of linear simultaneous inequations or inequalities which act as constraints.
The constraints are satisfied by any point which lies in (or sometimes) the boundary of any convex polygon. The coordinates of the accessible points provide a feasible solution to the linear programming problem. The main purpose of linear programming problem is ,given a convex polygon defined by a system of linear inequalities to find the points of the polygon at which a given linear expression where x and y are to be determined can take maximum or minimum values. If (x, y) point is restricted to lie in a closed convex polygon (the accessible region) then the linear expression assumes the maximum or minimum values at accessible
Bibliography
PBS Mathline: The Yo-Yo Problem (Solving Linear Equations): The High School Math Project- Focus On Algebra .http://www.pbs.org/Mathline. (Accessed November 19, 2012)
MEP: Demonstration project: Unit 16 Algebra: Linear Equation Activities. The Gatsby Charitable Foundation
Math Centre: Solving linear equations: MC-bus-lineqn-2009-1. www.mathcentre.ac.uk.
Ferguson, T.S: Linear programming: Concise introduction
Saylor, F & Sherri, N: Exploring Basic Linear Equations. http:// www.math.uakron.edu/eduPreAlgebraAlgebra//Linear Equations.
