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Example:Solving System of Equations Reducible to Linear Equation

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In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

since it makes all three equations valid.





Elimination of variables

The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:

1.  In the first equation, solve for the one of the variables in terms of the others.

2.  Plug this expression into the remaining equations. This yields a system of equations with one fewer equation and one fewer unknown.

3.  Continue until you have reduced the system to a single linear equation.

4. Solve this equation and then back-substitute until the entire solution is found. (Our solved example in mathguru.com uses this concept).


For example, consider the following system:

Solving the first equation for x gives x = 5 + 2z − 3y, and plugging this into the second and third equation yields

Solving the first of these equations for y yields y = 2 + 3z, and plugging this into the second equation yields z = 2. We now have:

Substituting z = 2 into the second equation gives y = 8, and substituting z = 2 and y = 8 into the first equation yields x = −15. Therefore, the solution set is the single point(x, y, z) = (−15, 8, 2).




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.