How to Solve Multivariable Linear Systems
Linear systems are sets of linear equations with multiple variables that can be solved together due to an interrelation. Linear systems already in row echelon form are easier to solve through back substitution of a known variable. More complex linear systems can be solved using Gaussian elimination, which employs the operations of switching equations, multiplying an equation by a nonzero number and then adding it to another equation to replace the latter equation.
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Instructions
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Row Echelon Form
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Use back substitution to find the solutions for the linear system including the following three equations: z - 2y + 3x = 6, y - z = -3 and z = 2. Note that one of your solutions, for "z", has been provided.
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2
Use back substitution to place the known value for "z" into the equation y - z = -3: y - 2 = -3. Add 2 to both sides to solve for "y": y = -1. Note that you now have two solutions: z = 2 and y = -1.
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3
Substitute your known values of "y" and "z" into the equation z - 2y + 3x = 6 or 2 - 2(-1) + 3x = 6 or 2 + 2 + 3x = 6 or 4 + 3x = 6. Subtract 4 from both sides: 3x = 2. Divide both sides by 3 to solve for "x": x = 3/2.
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4
Write the solution set as { ( 3/2 , -1 , 2 ) }.
Gaussian Elimination
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Use the methods of Gaussian Elimination to solve the linear equation set containing these two equations: 4x - 2y + z = 2 and y + 3z = 6.
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Solve the second equation for "y" by subtracting "3z" from both sides: y = -3z + 6. Note that because the answer contains "z," you have found the solution to "y" in terms of "z."
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7
Substitute this found "y" value into the equation 4x - 2y + z = 2 or 4x - 2(-3z + 6) + z = 2. Simplify the equation: 4x + 6z - 12 + z = 2 or 4x + 7z -12 = 2. Add 12 to both sides: 4x + 7z = 14. Subtract 7z from both sides to begin solving for "x": 4x = -7z + 14. Divide both sides by 4 to solve for "x": x = (-7z + 14)/4. Note that "x" has been solved in terms of "z."
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8
Simplify the solution set of equations into row echelon form by setting "z" equal to a new variable then replace the instances of "z" in the other answer equations. Using z = a, the other equations become y = -3a + 6 and x = (-7a + 14)/4.
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Tips & Warnings
Linear systems can have unique solutions, as displayed in the row echelon example, or multiple (parametric) solutions, which are necessary when there are more variables than there are equations, such as in the Gaussian example. It is also possible that solving a set will prove the set to be false, or unsolvable, in which case there is no solution.
