Why you can add equations using the Addition Method to solve a system of equations

For the longest time, I never understood how the Addition Method solved a system of simultaneous equations. What I mean to say is that I could carry out the method’s steps, but I didn’t understand why adding two equations was legal. It seemed unintuitive to me.

For example, take this system of two linear equations:

Untitled

Here is how I would solve it using the Addition Method:

procedure

Which is great, but why Step 3? I could understand the motivation for adding two equations to eliminate a variable, but what mathematical principle said it was okay to do that in the first place? It seemed to just about as rational to me as doing a head-stand while holding my nose to get the answer!

So I thought about it obsessively for a couple of hours and eventually realized the following:

The initial system basically says that y-x is 2 and y+x is 4. Conversely, 2 is y-x and 4 is y+x.

Nothing to write home about, but let’s say I want to add 2 and 4. Yes, the answer is 6, but suppose I want the answer to be in terms of x and y.

addmethhow1x

addmethhow2x

addmethhow3x

addmethhow4x

That is, Step 3! Effectively, we can add up the equations because a side of an equation can stand-in for its equivalent on the other side. The adding of equations is effectively the addition of numbers on the righthand sides of the two equations as the sum of those same numbers in terms of x and y.

For some, perhaps, this entry is a useless explanation of the obvious. Maybe it is, but no matter: now I understand something which I didn’t before, which I think is better than blindly accepting something as true, even if it’s small thing such as adding equations.

Cubic Polynomial Word Problem: Jenny and the Magic Bean

box

Here is a PDF version of a math problem I recently created. You’ll have to know how to factor cubic polynomials, so you may want to brush up on that first, or use this problem as a refresher. Just click on the link below to open it up. If you get stuck, a full solution starts on the second page.*

Enjoy!

WProb_CubicPolynomial_JennyMagicBean.pdf

* I know I’m being lazy here not writing the problem directly in the post, but it would take at least an hour’s work to insert all the LaTeX bits and to align things. Most browsers come with a PDF viewer, right ;-)?

Kinky Adventures in LaTeX Land: The Sum and Product of Roots of a Quadratic Equation

?????????🐒🐶🐱🐳🐿🐥?????????

\equiv \newline

\pi r^2 
\equiv \newline \pi r^2

?????????🙈😬🙄😮?????????

One of the things I don’t like about blogging is how long it can take to put together a blog post, even after it has been written, edited and polished. Aside from all the formatting required to make an entry look the way I want (see above), the other big time-suck has been adding mathematical statements (equations, formulas, expressions, etc.) to my blog. While I love MathType for its ease-of-use, using it has meant I’ve had to suffer through 1) writing math statements in the MathType Editor, 2) inserting them into a Pages document, 3) copying the said insertion into Preview, 4) exporting the Preview file as a PNG,  5) uploading the PNG to WordPress, and, finally, 6) inserting the PNG into the appropriate blog post. This is a lot of steps for something as short as, say, the area of circle (again, see above).

As such, I was happy when, several weeks ago, I stumbled onto MathJax, a Javascript engine which renders  LaTeX  “code” into professional-looking math statements. I had, for a long time, known about TeX and LaTeX as way to create math documents, but had always been satisfied with MathType to get done what I needed done. After doing a bit more investigation and discovering that WordPress already supported LaTeX,  I decided to give it a shot by using a LaTeX snippet I found here and expanding it into an entry on the sum and product of quadratic roots below.

***

Given a quadratic equation in general form ax^2 + bx + c = 0, the equation’s roots are

x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\newline \newline

or

x_1 = {-b + \sqrt{b^2-4ac} \over 2a}\newline \newline x_2 = {-b - \sqrt{b^2-4ac} \over 2a}\newline \newline

Continue reading “Kinky Adventures in LaTeX Land: The Sum and Product of Roots of a Quadratic Equation”

To cross multiply or not cross multiply

When I was in Grade 7 our homeroom/math teacher told the class in a hushed voice, “Don’t cross multiply.” At the time I didn’t understand what he meant, nor did I likely understand what cross multiplication was. Ever since I have occasionally wondered why he would admonish students from using such a useful tool. Cross-multiplication can tell you whether two proportions are equivalent and get you out of some tricky algebraic situations involving fractions.

Showing that two proportions are equivalent

img01_equiprop

Continue reading “To cross multiply or not cross multiply”

Algebraic Expressions vs Algebraic Equations

Give the algebraic expression that represents the perimeter of the following polygon. 
001 Perimeter Diagram

Over the last five years working as a tutor I have seen many students struggle with a problem like the one above. To answer it we must remember that the perimeter of any shape can be found by adding up its side-lengths. With side-lengths that are numbers this is straightforward, but with side-lengths that have variables and numbers it can be tricky—you need to know how to identify like-terms and how to sum them correctly.

001 Perimeter Expression

Therefore, the perimeter of the polygon is 13x-34 units.

But getting an answer such as 13x-34 is not necessarily what students have in mind as an answer. Often, they expect and, perhaps, want a number. Continue reading “Algebraic Expressions vs Algebraic Equations”