Systems of Equations

Since the previous age problem represents a system with two equations and two unknowns, it is called a system in two variables. Typically, three methods are used for determining the solutions for a system in two variables, including graphical, substitution and elimination.

By graphing the lines formed by each of the linear equations in the system, the solution to the age problem could have been obtained. The coordinates of any point in which the graphs intersect, or meet, represent a solution to the system because they must satisfy both equations. From a graph of these equations, it is obvious that there is only one solution to the system. In general, straight lines on a coordinate system are related in only three ways. First, they can be parallel lines which never cross and thus represent an inconsistent system without a solution. Second, they can intersect at one point, as in the previous example, representing a consistent system with one solution. And third, they can coincide, or intersect at all points indicating a dependent system that has an infinite number of solutions. Although it can provide some useful information, the graphical method is often difficult to use because it usually provides us with only approximate values for the solution of a system of equations.

The methods of substitution and elimination by addition give results with a good degree of accuracy. The substitution method involves using one of the equations in the system to solve for one variable in terms of the other. This value is then substituted into the first equation and a solution is obtained. Applying this method to the system of equations in the age problem, we would first rearrange the equation 1 in terms of x so it would become x = 2y. This value for x could then be substituted into equation 2 which would become 2y - 3y = -4, or simply y = 4. The value for x is then obtained by substituting y = 4 into either equation.

Probably the most important method of solution of a system of equations is the elimination method because it can be used for higher order systems. The method of elimination by addition involves replacing systems of equations with simpler equations, called equivalent systems. Consider the system with the following equations; equation 1: x - y = 1; and equation 2: x + y = 5. By the method of elimination, one of the variables is eliminated by adding together both equations and obtaining a simpler form. Thus equation 1 + equation 2 results in the simpler equation 2x = 6 or x =3. This value is then put back into the first equation to get y = 2.

Often, it is necessary to multiply equations by other variables or numbers to use the method of elimination. This can be illustrated by the system represented by the following equations:

In this case, addition of the equations will not result in a single equation with a single variable. However, by multiplying both sides of equation 2 by -2, it is transformed into -2x - 4y = -20. Now, this equivalent equation can be added to the first equation to obtain the simple equation, -3y = - 18 or y = 6.