Difference between revisions of "Systems of Equations in Three Variables"

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==Examples==
 
==Examples==
 
* One solution: <math> x + y + z = 3 </math>, <math> 2x - 3y + z = 0 </math>, and <math> -5x - 5y + 23z = 13 </math>. <math> x + y + z = 3 \implies 5x + 5y + 5z = 15</math>. We can add this to the third equation to get <math> 28z = 28 </math>, which means z = 1. So, the first two equations can be rewritten as <math> x + y = 2 </math> and <math> 2x - 3y = -1 </math>. 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).
 
* One solution: <math> x + y + z = 3 </math>, <math> 2x - 3y + z = 0 </math>, and <math> -5x - 5y + 23z = 13 </math>. <math> x + y + z = 3 \implies 5x + 5y + 5z = 15</math>. We can add this to the third equation to get <math> 28z = 28 </math>, which means z = 1. So, the first two equations can be rewritten as <math> x + y = 2 </math> and <math> 2x - 3y = -1 </math>. 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).
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* No solutions: <math> x + y + z = 1 </math>, <math> x + y + z = 2 </math>, <math> x + y + z = 3 </math>. 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.
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* No solutions: <math> x + y + z = 1 </math> and <math> x + y + z = 5 </math>. 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.
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* Infinite solutions: <math> x + y = z </math>, <math> x + y = 2z </math>, <math> x + y = 4z </math>. x + y = 0 for all x and y such that y = -x. Since <math> z = 2z = 4z </math> 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, <math> (3, -3, 0), (-0.5, 0.5, 0), </math> and <math> (\pi, -\pi, 0)</math> are solutions of this system of equations).
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* Infinite solutions: <math> x + y = z </math> and <math> x + y = 2z </math>. x + y = 0 for all x and y such that y = -x. Since <math> z = 2z </math> 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, <math> (3, -3, 0), (-0.5, 0.5, 0), </math> and <math> (\pi, -\pi, 0)</math> are solutions of this system of equations).
  
 
==Resources==
 
==Resources==

Revision as of 11:02, 15 September 2021

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: and . 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: and . x + y = 0 for all x and y such that y = -x. Since 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, and are solutions of this system of equations).

Resources

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