Systems of Equations
Systems of Equations
Systems in three or more variables
In mathematics, systems of equations, sometimes called simultaneous equations, are a group of relationships between various unknown variables that can be expressed in terms of algebraic expressions. The solutions for a simple system of equation can be obtained by graphing, substitution, and elimination by addition. These methods became too cumbersome to be used for more complex systems, however, and a method involving matrices was developed in order to find just complex solutions. Systems of equations have played an important part in the development of business. Faster methods for solutions continue to be explored.
Unknowns and linear equations
Many times, mathematical problems involve relationships between two variables. For example, the distance that a car, moving at 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 that 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 that 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 that relate the unknowns. An example of a system of equations can be described by the following problem involving the ages of two people. Suppose, this year, 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 one lets 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,
(Eq. 1) x – 2y = 0
(Eq. 2) 2x – 3y =–4
The solution of this system of equations will be any ordered pair that makes both equations true. This system has only solution, the ordered pair of x = 8 and y = 4, and is thus called consistent.
Solutions of linear 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 that 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 one 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, one 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:
(Eq.1) 2x – y = 2
(Eq. 2) x + 2y = 10
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, – 3 y = – 18 or y = 6.
Systems in three or more variables
Systems of equations with more than two variables are possible. A linear equation in three variables could be represented by the equation ax + by + cz = k, where a, b, c, and k are constants and x, y, and z are variables. For these systems, the solution set would contain all the number triplets that make the equation true. To obtain the solution to any system of equations, the number of unknowns must be equal to the number of equations available. Thus, to solve a system in three variables, there must exist three different equations that relate the unknowns.
Consistent system —A set of equations whose solution set is represented by only one ordered pair.
Dependent system —A set of equations whose solution set has an infinite amount of ordered pairs.
Elimination —A method for solving systems of equations that involves combining equations and reducing them to a simpler form.
Graphical solution —A method for finding the solution to a system of equations that involves graphing the equations and determining the points of intersection.
Inconsistent system —A set of equations that does not have a solution.
Linear equation —An algebraic expression that relates two variables and whose graph is a line.
Matrix —A rectangular array of numbers written in brackets and used to find solutions for complex systems of equations.
Ordered pair —A pair of values that can represent variables in a system of equations.
Solution set —The set of all ordered pairs that make a system of equations true.
Substitution —A method of determining the solutions to a system of equation that involves defining one variable in terms of another and substituting it into one of the equations.
The methods for solving a system of equations in three variables is analogous to the methods used to solve a two variable system and include graphical, substitution, and elimination. It should be noted that the graphs of these systems are represented by geometric planes instead of lines. The solutions by substitution and elimination, though more complex, are similar to the two variable system counterparts.
For systems of equations with more than three equations and three unknowns, the methods of graphing and substitution are not practical for determining a solution. Solutions for these types of systems are determined by using a mathematical invention known as a matrix. A matrix is represented by a rectangle array of numbers written in brackets. Each number in a matrix is known as an element. Matrices are categorized by their number of rows and columns.
By letting the elements in a matrix represent the constants in a system of equation, values for the variables that solve the equations can be obtained.
Systems of equations have played an important part in the development of business, industry and the military since the time of World War II (1939–1945). In these fields, solutions for systems of equations are obtained using computers and a method of maximizing parameters of the system called linear programming.
See also Graphs and graphing; Solution of equation.
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