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Graphing Systems of Equations

Objective Understand the geometric interpretation of systems of two linear equations in two variables, and in solving them graphically.

This chapter provides a great opportunity for you to discover ideas about systems of equations. Make sure that you graph several systems by yourself, at least one of which has no solution, and at least one of which has infinitely many solutions.

Systems of Linear Equations

Key Idea

A pair of linear equations like A + O = 5 and 2A + 3O = 12 is called a system of linear equations. Its solution set is the set of all values of the two variables that satisfy both equations.

So, let us solve the following system of linear equations:

A + O = 5 and 2A + 3O = 12

That is to say, A and O must satisfy the equations simultaneously. A good way to understand the solution to a system of equations is to graph both equations on the same axes, as shown at the right.

What information can be found from this graph?

First, the set of points on the graph of A + O = 5 is the set of all points that satisfy that equation, or the set of all ordered pairs (A , O ) for which the equation holds true. In the same way, the graph of 2A + 3O = 12 is the set of all points for which that equation holds true. Therefore, the solution to the system of equations is the set of all ordered pairs (A , O ) for which both equations hold true. The solution to the system must correspond to a point that lies on both graphs, or that lies on the intersection of the graphs.

There is only one intersection point, a point that lies on both graphs. It is the point with coordinates (3, 2), or A = 3 and O = 2.

You should graph these lines and find the coordinates of the intersection point yourself.

Exercise

Solve graphically the system x + 2y = 1 and 2x + y = 5.

In this case also, there is only one solution.

Is there always only one solution to a system of linear equations?

Think about the problem graphically. A system of linear equations corresponds to a pair of straight lines. When there are two straight lines, there are three possibilities.

 (1) The lines intersect in exactly one point. (2) The lines are parallel, and do not intersect at all. (3) The lines are actually the same line. In this case, there is exactly one solution, corresponding to the point of intersection. In this case, there are no solutions, since the graphs do not meet. In this case, there are infinitely many solutions, since every point on one of the lines lies on both lines.

The following exercises provide practice in solving each of the three types of systems.

Exercises

Determine whether each system has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it.

 y = 2x + 4 y = -x - 3 x + 4y = -8 y = 2x + 1 y = x + 3 no solution (-3, 0) infinitely many solutions