Systems of Equations in Three Variables

See Systems of Equations in Two Variables for more information on systems of equations.

Examples

• One solution: $x+y+z=3$, $2x-3y+z=0$, and $-5x-5y+23z=13$. $x+y+z=3\implies 5x+5y+5z=15$. We can add this to the third equation to get $28z=28$, which means z = 1. So, the first two equations can be rewritten as $x+y=2$ and $2x-3y=-1$. Using substitution, elimination, or graphing, we can calculate that x = 1 and y = 1 with these two equations. Thus, the solution to the system is (x, y, z) = (1, 1, 1).
• No solutions: $x+y+z=1$, $x+y+z=2$, $x+y+z=3$. These equations represent three parallel planes, and there is no x, y, and z that satisfy all three equations simultaneously. So, this system has no solutions.
• Infinite solutions: $x+y=z$, $x+y=2z$, $x+y=4z$. x + y = 0 for all x and y such that y = -x. Since $z=2z=4z$ when z = 0, this system has an infinite number of solutions of the form (x, -x, 0) where x can be any real number (for example, $(3,-3,0),(-0.5,0.5,0),$ and $(\pi ,-\pi ,0)$ are solutions of this system of equations).

Resources

• Linear Systems with Three Variables, Paul's Online Notes (Lamar Math)
• Systems of Equations: Three Variables, Lumen Learning
• Using Elimination to Solve a 3-Variable System, patrickJMT
• Solving a System of Two Equations with Three Variables (Infinite Solutions), patrickJMT