# Equations Reducible To Linear Equations

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.

For example, consider the following system:

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

Solving the first of these equations for *y* yields *y* = 2 + 3*z*, 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).