Content - Solution Using Graphing

Any corner of a rectangular solid is an intersecting point of three faces. Looked at another way, a corner of a rectangular solid is also the intersecting point of a line, which is the intersecting line of two faces, and a third face.

View 1, 2, and 3.

(System of Linear Equations with Three Variables)

Solution Using Graphing

Just as in the section Solving a System of Linear Equations with Two Variables Using Graphing, the goal is to visualize and gain an intuitive understanding of when the system will have a single solution, no solution, or unlimited solutions. The following system of simultaneous linear equations will be used to begin our discussion.

	2x - y + z = 4	(1)
	2x - y + 2z = 4	(2)
	5x + 5y + 3z = 15	(3)

The solution of the above system is the set of values x, y, and z that can satisfy the above three equations at the same time. To find the solution, we will determine the following three sets of values:

set #1 -	those values that can satisfy the equation (1);
set #2 -	those that can satisfy both equation (1) and (2);
set #3 -	those that can satisfy all three equations.

Apparently, only the set #3 is the solution to the system of the above simultaneous linear equations (1) through (3).

As discussed in the section Analytic Geometry in Three Dimensions, the graph of 2x - y + z = 4 is a plane. Any point in this plane satisfies this equation. Therefore, any point in this plane is its solution.

Similarly, any point in the plane 2x - y + 2z = 4 is the solution of equation (2). As shown, the intersection of the planes (1) and (2) is a straight line. Since any point on this line is on both planes (1) and (2) at the same time, this point can satisfy both equations (1) and (2) at the same time. Hence, the solutions for the first two equations are those points that are on the intersection line of planes (1) and (2).

How can we find those points that can satisfy all three equations simultaneously? Apparently, if any point on the intersection line of planes (1) and (2) can satisfy equation (3), then this point will be the solution of the system, because this point satisfies all three equations at the same time. How do we find such a point? As shown by the graph, the intersection line of planes (1) and (2) intersects plane (3) at one point. Obviously, this point is on the intersection line and also in the plane (3). Actually, this point is the intersecting point of all three planes. Therefore, this point is the solution of the system.

What happens if the intersection line of planes (1) and (2) does not intersect plane (3)? This is exactly the case if we replace equation (3) with 2x - y + z = 6. Since we cannot find a point that can satisfy all three equations at the same time, the solution does not exist. This is because the intersecting line of (1) and (2) is parallel to the plane of (3). Therefore, no intersecting point exists. Hence, there is no solution.

When will there be more than one solution? The answer is when the intersection line of planes (1) and (2) intersects plane (3) at more than one location. How can this happen? This can occur when plane (3) overlaps with one of the first two planes or when plane (3) passes through the intersection line, as shown in the graph.

You may have realized that the graphing method is not always sufficient to obtain accurate numerical values of the solution. Its strength lies in its ability to help you visualize the conditions under which there will be a single solution, no solution, and unlimited numbers of solutions.

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