Systems of Equations in Three Variables

From Department of Mathematics at UTSA
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See Systems of Equations in Two Variables for more information on systems of equations.

Examples

  • One solution: , , and . . We can add this to the third equation to get , which means z = 1. So, the first two equations can be rewritten as and . 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: , , . 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 = 0 for all x and y such that y = -x. Since when z = 0, this system of equations has an infinite number of solutions along the line y = -x (for example, , etc.).

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