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Solving Systems in Three Variable (Lesson 6.4 Day 2)

Unit 6 - Day 6

​Learning Objectives​
  • Determine when a system in three variables results in an inconsistent or dependent system

  • Write the general solution to a dependent system of equations

Quick Lesson Plan
Activity: Gaussian Elimination Practice



Lesson Handout

Answer Key

Experience First

We started class today by looking at a dependent system (though I did not tell students this at the beginning). While we completed this problem in a whole-class setting, we used notice and wonder protocols and turn and talks to garner discussion about this particular system. Students suggested “moves” to make and then noticed that row 3 was exactly 3 times row 2. We had students make conjectures about what this means, thinking back to Kelly’s Panera Bread order. Students came up with the idea that this system was dependent but we wanted to verify this by doing the final “move” to show the row of zeros. This tied back to what students already knew about 2-variable systems. One way to make the connection between a dependent system and seeing a row of zeros is to ask students to explain what that row means. They will say 0x+0y+0z=0 or in other words 0=0. I ask them if this is always, sometimes, or never true. Since 0 is always zero, we conclude that the value of variable z is indeterminate and can be whatever it wants, or that it is a free variable.


The harder part for students is writing the general solution. After establishing that z is the free variable, we write z in our ordered triple solution. I then ask students “How is y related to z?” From row 2 we learn that y and z are the same, so whatever z is, y is also that value. We then write z for the y-coordinate of our ordered triple. A similar approach is taken to find x, though the relationship is not quite so simple. Re-writing equation 1 gives that x=-1+3z-y which becomes x=-1+2z. Ask several students: “What if z=8, what would y be? What would x be?” and change the value of z each time. Ask students what picture from page 1 of yesterday’s lesson is represented by this system.


The final part of this class discussion asks students to think about what the system would look like if there were no solutions instead of infinitely many. Students are able to apply their understanding from Lesson 6.3 and conclude that there would be a row of zeros equaling some number that is not zero. Again, I ask them “is this always, sometimes, or never true?” When students respond with “never” we can conclude that the system is inconsistent.

Formalize Later

We have found that students need adequate time to practice Gaussian elimination. The rest of the class is spent having students work with a partner or group of three at a whiteboard working through 3-5 Gaussian elimination problems. When choosing practice problems, try them yourself first, as many of them require more steps than students may be ready for at this point. We suggest having at least one of the assigned problems feature a dependent system, and one problem feature an inconsistent system. Always have students write the general solution for a dependent system.


Allow students to be “inefficient” and take the long way to row-echelon form. Students need time to develop efficient strategies, and offering our suggestions and shortcuts, while they may save time, also deprive students of some of the productive struggle needed to build fluency over time.

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