Question

Two Column "Proof" Tutorial

Answer 1

So as many people seem to have issues with these, I thought I might write a tutorial on how to go about filling in the blanks of this type of proof. Now, my personal preference is to write this out in a paragraph, but, that is not the standard two column proof, so without further ado. Let us discuss this common geometry and algebra topic.

Answer 2

Let us first examine the column headings

Answer 3

As you see there are two, hence the "Two-column proof". But what is important, is to think about what will be in the columns . So the "steps" column will host every single itty bitty step that you do by number. The reasons column will tell the reader why you are able to make that step occur.

Answer 4

Now here is our example problem. Our first step is always our problem that the book/teacher/notes/assignment, have given us. We want to rewrite this as the first line of our steps, so our reason for that very first line, will always be "given."

Answer 5

So we start our problem with
\[-(x+y)+x=-y \]
Our first step to simplify this is to get rid of those parentheses. In order to do this, we use the Distributive property to obtain:
\[-x+(-y)+x = -y\] we can also write this as \[-x-y+x= -y\]
Either way is fine, but I've been seeing the former in the questions.


So since in order to go from step one to step two we used the distributive property, that is our justification/reason number two.

\[\color\red{\text{Please note that for each reason item,} \\ \text{
they will always refer to how you went between the step} \\ \text{
previous and the step you are on. IE if you are at reason} \\ \text{
number 3, then that reason tells you how you went from
step 2 to step 3.}}\]

Answer 6

Our next step, is to move our like terms together using the Commutative property. So we go from \[−x+(−y)+x=−y\]to\[-x+x+(-y) =-y\]

Answer 7

Can you figure out what step 4 will be? Let's see if we can figure out the step based on the justification.

Answer 8

So for this part, we are told that the justification for how we went form step 3 to step 4 is the additive inverse property. So let us think, what does the additive inverse property state? a+(-a)=0 or a-a=0. Alright, so where can we apply this in \[−x+x+(−y)=−y?\]

Answer 9

If you noticed \[\color\red{−x+x}+(−y)=−y\] then you have indeed found the step. So applying the additive inverse to that highlighted in red, we obtain: \[0+(−y)=−y\]
And this, is our step number 4.

Answer 10

Can you figure out step 5 by the reason?