Systems of Equations

Many times, mathematical problems involve relationships between two variables. For example, the distance that a car moving 55 mph travels in a unit of time can be described by the equation y = 55x. In this case, y is the distance traveled, x is the time and the equation is known as a linear equation in two variables. Note that for every value of x, there is a value of y which makes the equation true. For instance, when x is 1, y is 55. Similarly, when x is 4, y is 220. Any pair of values, or ordered pair, which make the equation true are known as the solution of the equation. The set of all ordered pairs which make the equation true are called the solution set. Linear equations are more generally written as ax + by = c where a, b and c represent constants and x and y represent unknowns.

Often, two unknowns can be related to each other by more than one equation. A system of equations includes all of the linear equations which relate the unknowns. An example of a system of equations can be described by the following problem involving the ages of two people. Suppose Lynn is twice as old as Ruthie, but two years ago, Lynn was three times as old as Ruthie. Two equations can be written for this problem. If we let x = Lynn's age and y = Ruthie's age, then the two equations relating the unknown ages would be x = 2y and x - 2 = 3(y - 2). The relationships can be rewritten in the general format for linear equations to obtain,

The solution of this system of equations will be any ordered pair which makes both equations true. This system has only solution, the ordered pair of x = 8 and y = 4, and is thus called consistent.