Systems of Equations in Three Variables

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

Examples

One solution: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+y+z=3} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2x-3y+z=0} , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -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: $x+y+z=1$ , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+y+z=2} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+y=z} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+y=2z} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+y=4z} . x + y = 0 for all x and y such that y = -x. Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z=2z=4z} when z = 0, this system of equations has an infinite number of solutions of the form (x, -x, 0) (for example, $(3,-3,0),(-2,2,0),(\pi ,-\pi ,0)$ , etc.).