Systems of 3×3 Equations interactive applet

In the Graphical Solutions for Linear Systems page in the earlier Systems of Equations chapter, we learned that the solution of a 2×2 system of equations can be represented by the intersection point of the two straight lines representing the two given equations.

We extend that idea here to systems of 3×3 equations (that is, 3 equations in 3 unknowns).

Graphical representation of a 3×3 system of equations

The following applet demonstrates what we are doing when we solve a set of 3 simultaneous equations.

As an example, consider the following 3 simultaneous equations:

2x + 3y + z = 2

x + 2y + 3z = −1

−3x − 3y + z = 0

Each equation represents a plane in 3 dimensions, and the point of intersection, P(4, −3, 3), is the point common to all 3 planes. (Try substituting that point in the equations for the 3 planes. It works each time. We will learn how to find this point of intersection later in this chapter.)

You can choose two other sets of simultaneous equations (which are actually examples given eslewhere in this chapter) near the top of the applet.

The applet

Things to do

You can:

  • Toggle (show or hide) each of the 3 planes and the intersect point, P, so you can see what "3 planes meeting at a point" means.
  • Drag the graph left-right and up-down to see the planes and point P more clearly from different angles
  • Choose a different set of equations. Each of these sets comes from examples in this chapter
  • You can zoom in and out using the mouse wheel (or pinching, if on a mobile device)
  • You can pan the whole graph left, right, up or down using the right mouse button and dragging

Choose equations set:

System 1 System 2 System 3


2x + 3y + z = 2

x + 2y + 3z = −1

−3x − 3y + z = 0

Intersection point

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Next, we introduce matrices, which is the way we'll solve such systems of equations:

3. Matrices