2.1 A Standard Classroom Example

Consider the following problem, typical of the problems found in a standard high school algebra text:

The sum of two numbers is 400. Three times the first, minus seven times the second is 200. What are the numbers?

 

	x	+	y	=	400
	3x	-	7y	=	200	(2.1)

The two equations in (2.1) are linear (the exponent of each variable is one, and there are no combinations of variables other than by addition or subtraction). Together, the two equations form a system of linear equations. A combination of values of x and y that makes both equations true is called a solution of this system of linear equations. If there is only one combination of values of x and y that satisfies both equations, then these values are said to form a unique solution of the system.

Using a simple method, such as substitution or elimination or even "guess and check", it is easy to determine that x = 300 and y = 100.

You have probably seen this type of problem in a textbook. It does not model any meaningful system; it is just a contrived problem that someone made up to give you a chance to practice solving systems of linear equations by hand. It is not the type of problem that computational scientists are asked to solve. However, computational scientists do deal frequently with systems of linear equations -- either as models of real-world problems or as a part of the solution of more complicated problems. A more realistic problem that can be modeled with a simple linear system is presented in the next section.