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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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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 Cramerâ€™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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