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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y + z = 3 } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: $x+y+z=1$ and $x+y+z=5$ . These equations represent two parallel planes, and there is no x, y, and z that satisfy both equations simultaneously. So, this system has no solutions.
Infinite solutions: $x+y=z$ and $x+y=2z$ . x + y = 0 for all x and y such that y = -x. Since $z=2z$ 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\pi, -\pi, 0)} are solutions of this system of equations).