# Calling for conceptual help!

7 Oct

I know I’m a nerd when a question like the following is my hot topic of the day:

Why is it OK to replace a row in a matrix with the answer from adding two rows?”

For example, in solving for:

x + y + z = 0
x – y + z = 2
x – y – z = 10

a matrix like this can be used:

1   1   1   0
1  -1   1   2
1  -1  -1  10

The next step would be to add two of the rows and replace one of the equations

1   1   1   0
2  0   2   2
1  -1  -1  10

But WHY is this allowed? It’s not like I’m replacing the row with an equivalent equation…

After some discussion and thought, a coworker and I came upon one conceptual theory. Conceptually, what we are doing is replacing an equation of a line with a new equation that got rid of one (or two) of the variables. Each of the variables, in this case, represent an added dimension.

Therefore, what I think we are doing when replacing a row is that we’re getting rid of dimensions and using a model in a lower dimension to represent the original line.

For example, this is how I see it:
Picture a line sticking diagonally out from your computer screen. This line is definitely in 3D. Now imagine taking a picture of that line coming out of the screen. This picture on your camera now made a 3D line into a new 2D line!

It’s kind of like taking a 3D object and taking a picture of that object. That picture is a 2-dimensional representation of the 3D object. This object was ORIGINALLY in 3-dimensions Taking a picture of it, however, makes this new representation of the same cube, 2D

I think the same thing is happening with these lines. When solving a system of equations like

x + y + z = 0
x – y + z = 2
x – y – z = 10

what we are doing when looking for a “solution” is finding the point of intersection of the 3 lines. So in order to do this, I think we’re taking “pictures” of the lines to simplify our 3D model to 2D and then to a 1D representation that reveals the point of intersection.

So again, this is only what I’ve theorized so far. I can picture it and I’m looking for a way to visually prove it. I’ve never actually taught this before in Algebra 2, always due to lack of time. I finally decided to teach it this year and would love any kind of feedback!

QUESTIONS:

1. Does this mean I can replace a row only with a new equation that gets rid of a variable/ dimension? What does it mean if I replace a row with a combined equation that does NOT get rid of a variable? Is this like taking a picture from a different angle not parallel to one of the axis?
2. If all this is true, why does COMBINING the two equations give you an accurate “picture” of the 3D model?

### 4 Responses to “Calling for conceptual help!”

1. Aileen October 8, 2009 at 2:17 am #

Math = Jinna = Sexy. J’adore math!

2. Jessica October 8, 2009 at 3:03 am #

This isn’t directly related to this post, but more a tangent off of a comment I saw you made at Sweeney Math re: the Quadratic Formula Rap. You said you use sign while singing Quadratic Formula in “pop goes the weasel”. Do you teach deaf/hard-of-hearing students? I do…I’m looking to network with other math teachers of the deaf. I’m teaching calculus for the first time this year…pretty exciting!

3. talkingninja October 9, 2009 at 4:36 pm #

Hi Jessica,
I actually don’t teach any deaf or hard of hearing. A colleague of mine taught me the formula and it turned out that most of the kids loved the change and even the challenge of signing for the first time. If you have any other suggestions for a math lesson in sign language I would love to try it! (I have a couple deaf studies friends who can help show me the actual signs)
4. 