Systems of Three Linear Equations – Planes and Gaussian Elimination
MathematicsAlgebraAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Algebra
Each equation ax + by + cz = d is a plane in space. Rotate the 3D frame to see three planes meet at one point (unique solution), meet pairwise along three parallel lines (no solution), or share a common line (infinite solutions), while stepping through Gaussian elimination on the augmented matrix to see why the system falls into that case. The 2D | 3D button switches to a true 3D view (WebGL): the three colored planes in the Oxyz axes, their lines of intersection and the solution point, free to rotate and zoom.
Lesson: Systems of three linear equations
What it shows
A linear equation ax + by + cz = d in three unknowns describes a plane in space, so solving a system of three such equations means finding the points common to three planes. The planes can meet at a single point (unique solution), share a common line (infinitely many solutions with one free variable), or have no common point, for example when they meet pairwise in three parallel lines. Gaussian elimination on the augmented matrix reaches the same conclusion algebraically: a row reading 0 = a nonzero number means no solution, while an all-zero row means one equation depends on the others.
How to use
Choose a System and ask students to predict how the three planes are arranged. Rotate the figure, or use the 2D | 3D button for the WebGL view, to check the prediction. Then click Next step repeatedly and relate each row operation to the matrix, going back with Previous step when needed. For the System with parameter m, drag Parameter m and find the value at which the unique solution disappears.
Parameters you can change
- System Shopping problem (unique solution), System with parameter m, No solution, Infinite solutions
- Parameter m (used for the parametric system) -6–6
- Currency (shopping problem) From the language, US dollar, Euro, Indian rupee, Indonesian rupiah, Saudi riyal
Questions to explore
- If three planes are parallel to one another, can the system have a solution? What would elimination show?
- For which value of m does the parametric system have no solution, and which row appears after elimination?
- Why does a system with infinitely many solutions correspond to a line shared by all three planes?
