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Solving Systems in Three Variables (Lesson 6.4 Day 1)

Unit 6 - Day 5

​Learning Objectives​
  • Explain the importance of row-echelon form in solving a three variable linear system

  • Solve independent linear systems in three variables using Gaussian elimination

Quick Lesson Plan
Activity: The Matrix Game



Lesson Handout

Answer Key

Experience First

This activity begins with a visual representation of systems in three variables. Students learn that linear equations in 3 variables represent planes instead of lines and that there are many ways for systems to be inconsistent. We suggest having groups just do question 1, then discussing as a whole class before moving through the rest of the activity.


The first half of the lesson is designed to demonstrate the value of row-echelon form. Students identify these forms as being “easy to solve” since they already give the value of one variable and then only require back-substituting. The second half of the lesson has students consider what “moves” they might make (using the idea of equivalence) to get a matrix into row-echelon form.


Although there are multiple ways to teach Gaussian elimination, we have found that using matrices helps students better organize their work, even though they have little to no past experience using matrices. Question 3 has students identify the structure of a matrix and its potential value.


We use the idea of a game to emphasize to students that there is no one correct procedure for arriving at row-echelon form. We want students to be strategic in their choices and over time become more efficient in their methods. We often have students articulate why they decided to make a particular “move”. 

Formalize Later

In this lesson new notation is introduced and important ideas about equivalence are reinforced. It is worth discussing with students why we can switch rows, or multiply by a constant (think back to Kelly’s Panera order!). Students practice eliminating variables (by getting 0s underneath the main diagonal) and substituting (once the matrix is in row-echelon form). These ideas themselves are not new, but putting everything together is, so we only focus on independent linear systems on day 1. Tomorrow we will look at what happens when three planes never intersect or when they intersect at a line (infinitely many solutions).

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