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Since we’re now working with three planes instead of two lines, we have more scenarios that can result in infinite or no solution. Just like with two variable systems of equations you can have infinitely many solutions or no solution. Again, this is easier in practice so check out the tutorial below to see it in action :)įor more help with Gaussian Elimination check out this tutorial with step-by-step instructions. This ensures that the two resulting two-variable equations are capable of being solved using the substitution or elimination methods. The key to completing the backsolving method successfully is to eliminate the same variable from your two sets of three-variable equations.
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#STEPS FOR 3 EQUATION SYSTEMS HOW TO#
In the following video tutorial, I demonstrate how to choose two sets of two equations to perform the elimination method on, and then back solve to solve for all three variables. It’s a bit tedious, but a fairly simple process once you get the hang of it. Once you find one variable’s solution, you can then back solve to find the solutions for the other two variables. The basic idea behind the backsolving method is to take your 3 three-variable equations and make 2 two-variable equations so that you can solve them using the traditional substitution or elimination methods. In fact, you’ll be using the Elimination method multiple times in the course of the backsolving method. The Backsolving Method is reminiscent of the Elimination Method from our standard systems of equations arsenal. If you are in a Linear Algebra or computer programming course, you’ll probably be asked to solve using Gaussian Elimination with matrices. If you are in second-year Algebra or PreCalculus, most likely you’ll be using the Backsolving Method. The main difference is whether you want or need to solve the system using equations or matrices. The two main methods are called the Backsolving Method and Gaussian Elimination/Row Reduction Method (sometimes called Gauss-Jordan Elimination). Just like when you were solving two-variable systems of equations you had multiple methods to choose from, you have a couple of options here as well. In some cases, you may be able to solve a three-variable system of equations with only two equations, but it isn’t as common. This is similar to how you need two equations to solve a standard system of linear equations. In general, you’ll be given three equations to solve a three-variable system of equations. It’s probably sufficient for you to understand that the solution to your three-variable system of equations is the values that will make all three equations true and represents the intersection of three planes in 3D space. Don’t worry, you seldom will be asked to graph a set of three variable systems in 3D space. Visually, a three-variable equation is represented in 3-dimensional space as a plane in the xyz-coordinate axis. The solution still represents the values for x, y, and z (or whatever variables your equations are using) that when plugged into each equation holds true.
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Three variable systems of equations aren’t so different. In other words, the solution is the value or values for x and y that hold true for both equations.
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The first question you may have is what exactly is this three-variable system? You may remember from two-variable systems of equations, the equations each represent a line on an XY-coordinate plane, and the solution is the (x,y) intersection point for the two lines. What is a Three-Variable System of Equations?