Systems of Equations

We have worked with two types of equations--equations with one variable and equations with two variables. In general, we could find a limited number of solutions to a single equation with one variable, while we could find an infinite number of solutions to a single equation with two variables. This is because a single equation with two variables is underdetermined--there are more variables than equations. But what if we added another equation?

A system of equations is a set of two or more equations with the same variables. A solution to a system of equations is a set of values for the variable that satisfy all the equations simultaneously. In order to solve a system of equations, one must find all the sets of values of the variables that constitutes solutions of the system.

Example: Which of the ordered pairs in the set {(5, 4),(3, 8),(6, 4),(4, 6),(7, 2)} is a solution of the following system of equations:



(5, 4) is a solution of the first equation, but not the second.
(3, 8) is a solution of both equations.
(6, 4) is a solution of the second equation, but not the first.
(4, 6) is a solution of both equations.
(7, 2) is not a solution of either equation.
Thus, the solution set of the system is {(3, 8),(4, 6)}.

When we graph a linear equation in two variables as a line in the plane, all the points on this line correspond to ordered pairs that satisfy the equation. Thus, when we graph two equations, all the points of intersection--the points which lie on both lines--are the points which satisfy both equations.

To solve a system of equations by graphing, graph all the equations in the system. The point(s) at which all the lines intersect are the solutions to the system.

Example: Solve the following system by graphing:



Since the two lines intersect at the point (1, 1), this point is a solution to the system. Thus, the solution set to the system of equations is {(1, 1)}.

To check, plug (1, 1) in to both equations:
1 - 3 = - (1 + 2) ? Yes.
1 = 3(1) - 2 ? Yes.

There are three possibilities for the manner in which the graphs of two linear equations could meet--the lines could intersect once, not intersect at all (be parallel), or intersect an infinite number of times (in which case the two lines are actually the same).

If the two equations describe the same line, and thus lines that intersect an infinite number of times, the system is dependent and consistent.

If the two equations describe lines that intersect once, the system is independent and consistent.

If the two equations describe parallel lines, and thus lines that do not intersect, the system is independent and inconsistent.

Thus, a system is consistent if it has one or more solutions. A system of two equations is dependent if all solutions to one equation are also solutions to the other equation.

The following chart will help determine if an equation is consistent and if an equation is dependent: