Equations Reducible To Linear Equations

 

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 xyz. 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.

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(xyz) = (−15, 8, 2).