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Solving Systems of Linear Equations in Three Variables

In Maths it is many times necessary to solve systems of linear equations, such as

x + 2y + 3z = 1
2x + 5y + 2z = 2
x + y + z = 3

There are are at least three ways to solve this set of equations: Elimination of variables, Gaussian reduction, and Cramer’s rule. The first approach is described below.

 

Elimination of Variables

In the example, you first eliminate x from the second two equations, by subtracting twice the first equation from the second, and subtracting the first equation from the third. The three equations then become

x + 2y + 3z = 1
    y - 4z = 0
    -y - 2z = 2

Next, y is eliminated from the third equation, by adding the (new) second equation to the third, yielding

x + 2y + 3z = 1
    y - 4z = 0
      - 6z = 2

From the third equation, we conclude that

From the second equation, we conclude that

Finally, from the first equation, we find that

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