Determine the number of solutions to a system of equations in three variables

Learning Objectives Differentiate between consistent and inconsistent systems of equations in three variables. Identify if a system has one unique solution, no solution, or infinitely many solutions. Use the elimination or substitution method to reduce a 3x3 system to a 2x2 system. Interpret the algebraic results of the elimination process (e.g., 0 = 5 vs. 0 = 0). Describe the geometric interpretation of each solution type as the intersection of planes in 3D space. Recognize the algebraic indicators of dependent and inconsistent systems without fully solving for a variable. How does your phone's GPS pinpoint your exact location in 3D space? 🛰️ It uses a system of equations where the solution is your unique position! In this tutorial, we'll move beyond just finding...

TermDefinitionExample System of Linear Equations in Three VariablesA set of three linear equations, each containing up to three variables (typically x, y, and z). Geometrically, each equation represents a plane in three-dimensional space.1) x + 2y - z = 4 2) 2x - y + 3z = 9 3) -x + 3y + z = 6 Consistent SystemA system of equations that has at least one solution. The planes intersect at one or more points.A system with a single solution like (1, 2, 1) or a system with infinite solutions are both considered consistent. Inconsistent SystemA system of equations that has no solution. Geometrically, this means the three planes never intersect at a common point (e.g., they are parallel or intersect in pairs but not all three together).Solving the system leads to a contradiction, such as 0 = 8. D...

The Contradiction Rule (No Solution) If the process of elimination results in a false statement, the system is inconsistent. After combining equations, if you arrive at an equation with no variables that is mathematically false (e.g., 0 = c, where c is a non-zero constant), the system has no solution. The planes never share a common intersection point. The Identity Rule (Infinite Solutions) If the process of elimination results in a true statement (an identity), the system is dependent. If you arrive at an equation with no variables that is always true (e.g., 0 = 0 or c = c), the system has infinitely many solutions. This indicates that the equations are not independent of each other. The planes intersect along a line or are the same plane. The Unique Value Rule (One Sol...