Algebraic Equivalence

Algebraic Equivalence - Mental Model Series - 11

“It's only in Algebra that two negatives make a positive” ― Charmaine J Forde

Algebraic equivalence teaches us that two things need not be the same in order to be equal. Let us spend time on understanding this term how it can be relevant and useful to decision making

Firstly ,Equivalent expressions are essentially just different expressions that will yield the same answer. For example:
10 + 10 = 20
5 x 4 = 20
40/2 = 20

If you apply symbols to these numbers on the left side and pretend that we don't them , this is how those expressions will look like ,
 x+ x =  20, in other words  2X =20
 Y x4   =20  , in other words  4Y  = 20
 Z/2 =20

So the value of 20 can be deduced by

2X or 4Y or Z/2

Also these symbols are used to represent unknown numbers that can be solved for given other relevant information(20). The general point about algebraic equivalence is that it teaches us that two things need not be the same in order to be equal.

Equivalent equations in daily life:

 It's particularly helpful when shopping. For example, you like a particular shirt. One company offers the shirt for $6 and has $12 shipping, while another company offers the shirt for $7.50 and has $9 shipping. Which shirt has the best price? How many shirts (maybe you want to get them for friends) would you have to buy for the price to be the same for both companies?

To solve this problem, let "x" be the number of shirts. To start with, set x =1 for the purchase of one shirt.

For company #1:

Price = 6x + 12 = (6)(1) + 12 = 6 + 12 = $18

For company #2:

Price = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.5

So, if you're buying one shirt, the second company offers a better deal.

To find the point where prices are equal, let "x" remain the number of shirts, but set the two equations equal to each other. Solve for "x" to find how many shirts you'd have to buy:

6x + 12 = 7.5x + 9

6x - 7.5x = 9 - 12 (subtracting the same numbers or expressions from each side)

-1.5x = -3

1.5x = 3 (dividing both sides by the same number, -1)

x = 3/1.5 (dividing both sides by 1.5)

x = 2

If you buy two shirts, the price is the same, no matter where you get it. You can use the same math to determine which company gives you a better deal with larger orders and also to calculate how much you'll save using one company over another.

Let me quote another excellent example from Farnam street blog on this topic :

In a deeper way, algebraic equivalence helps us deal with one accusation that all parents get at one time or another: “You love my sibling more than me.” It’s not true, but our default usually is to say, “No, I love you both the same.” This can be confusing for children, because, after all, they are not the same as their sibling, and you likely interact with them differently, so how can the love be the same?
Using algebraic equivalence as a model shifts it. You can respond instead that you love them both equally. Even though what’s on either side of the equation is different, it is equal. Swinging the younger child up in the air is equivalent to asking the older one about her school project. Appreciating one’s sense of humor is equivalent to respecting the other’s organizational abilities. They may be different, but the love is equal.

Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 
For More , read https://jamesclear.com/feynman-mental-models