Solving Linear Systems in Three Variables

In this section, we will learn how to solve a linear system when three variables are involved. Since we are now dealing with three variables instead of two, our solution will change from an ordered pair: (x,y), to an ordered triple: (x,y,z). To solve a linear system with three variables, we begin by eliminating a variable from any two of the equations. Next, we eliminate the same variable from a different pair of equations. This will lead to a two-variable system with two equations. We can then solve this system and find two of our unknowns. Lastly, we can plug into one of our original equations to find the third and final unknown. Always be sure to check the ordered triple: (x, y, z) in each equation of the system.