Systems of Linear Equations

Systems of three equations

Use elimination (row reduction) or the GDC.

Example.

x + y + z = 6

2x − y + z = 3

x + 2y − z = 2

Solution: x = 1, y = 2, z = 3.

Three possible outcomes

Outcome	Meaning geometrically (three planes)
unique solution	the planes meet at one point
infinitely many solutions	the planes share a line (or are the same plane)
no solution	no common point (parallel planes or a triangular prism)

When elimination gives 0 = 0, there are infinitely many solutions; when it gives 0 = c with c ≠ 0, there are none.

Parametric solutions

If there are infinitely many solutions, write them with a parameter.

x + y + z = 3 and x − y + z = 1 give y = 1 and x + z = 2: x = 2 − λ, y = 1, z = λ.

Systems with an unknown constant

For which k does a system have no solution? Reduce it and find the k that produces 0 = nonzero.

Quick summary

• Eliminate one variable at a time, or use the GDC.
• 0 = 0: infinitely many solutions; 0 = c: no solution.
• Express infinite solution sets with a parameter.