No solution Two Equations Containing Two Variables The first two cases are called consistent since there are solutions. This makes solving a system of equations in three variables by graphing rather difficult. The three planes can appear in various configurations. For a system of equations in three variables, the graph of each equation is a plane in space rather than a line. A dependent system with infinitely many solutions 3. You used graphing to solve a system of equations in two variables. Exactly one solution, an ordered pair (x, y) 2. Systems of Linear Equations The solution will be one of three cases: 1. Multiply both sides of an equation by a nonzero constant. Solving systems of equations in 3 variables. The outcomes for these systems of equations are: 1.) An intersection at one point (an ordered triple). The best way to imagine this is to think of the point as a corner of a box. To find a solution, we can perform the following operations: Interchange the order of any two equations. In systems of linear equations in three variables the desired solution is an ordered triple (x, y, z) that exists in three-dimensional space. Therefore, in the end, you will have successfully have found the answers to a system of linear equations in three variables. A solution to a system of three equations in three variables (x,y,z), ( x, y, z), is called an ordered triple.After solving for another variable, you should have the remaining pieces of the puzzle for the last equation.Note: Preferably, plug in the value to the most simplified equation. After solving for the final variable, plug in the value of the most recent variable that you found (in terms of the example, y=3) into the answer of another equations with variables remaining (in terms of the example, z = y � 6, x = 6).To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. This step should allow you to solve for a real number. 4.4.2 Solve a System of Linear Equations with Three Variables. Substitute the value from the two variables that you solved and plug it into the remaining equation and solve for the last remaining variable.Next, substitute the value from the first variable you solved for into the other equation and solve for the next variable.From the three variables, there is no incorrect choice so choose to solve for any variable. Solve one of the equations for one of its variables.Steps in order to solve systems of linear equations through substitution: This will be the sample equation used through out the instructions: The substitution method involves algebraic substitution of one equation into a variable of the other. Solving a Linear System of Linear Equations in Three Variables by Substitution