In this Section:
In this section, we expand our knowledge and learn about solving systems of linear equations in three variables. These problems will deal with three equations and three variables x, y, and z. When we encounter a linear equation in three variables, the solution is written as an ‘ordered triple’ (x,y,z). There are many methods that can be used to solve a system of linear equations in three variables. Most texts will suggest an extension of the elimination method. With this method, we choose one of the variables to eliminate. We eliminate this variable from any two equations of the system. We then eliminate the same variable from any two other equations. This produces a linear system in two variables. From this point, we can solve using any of the techniques we know: elimination or substitution. Once we solve the linear system in two variables, we plug the two known values into one of the original equations and solve for the final unknown. As we usually do, we check by plugging into each original equation. We will also encounter systems with no solution, or infinitely many solutions.