The way Main nun thought converting numbers into duckies will make it easier on her life, which it obviously didn't, reminds me of a certain comedy short.
In a certain rural village, lived a certain nomad who is very, very smart. Ask him anything and he can give you an answer almost instantly. One day, a farmer from the village came to him with a problem. He has 9 hens which lay eggs every 20 days, this morning he bough 12 more and so he asked: now how long will it takes for the hens to lay eggs.
The answer is obviously still 20 days because they're still chicken. Increasing their numbers doesn't just turn them into a whole different species with new egg-laying cycle.
But the nomad, somehow, converted the whole thing into a math equation with the cycle being X, and spent the rest of his life trying to solve it.
My god. I feel old. I know what x is, but I can't recall for the life of me on the formula to get x.
Just one by one move all non-X value to the right side of "=" and change their arithmetic with the opposite one (addition for subtraction, multiplication for division and vice versa)
Just one by one move all non-X value to the right side of "=" and change their arithmetic with the opposite one (addition for subtraction, multiplication for division and vice versa)
So, putting my brain to work. I move 3 to 15 and subtract it, so that's 2x=12, then I divide both by 2 so that it becomes x=6, yeah?
Man, it's been a long time. I remember doing harder stuff without a calculator, and now my brain is mush.
So, putting my brain to work. I move 3 to 15 and subtract it, so that's 2x=12, then I divide both by 2 so that it becomes x=6, yeah?
Man, it's been a long time. I remember doing harder stuff without a calculator, and now my brain is mush.
Yes.
The key thing to note is that you can apply the same operations to both sides while still preserving equality. So the goal of "solve for x" (single variable) is to run identical operations on both sides until you get a single variable x on the left side and a number on the right. So, basically:
2x + 3 = 15
Step 1. 2x + 3 - 3 = 15 - 3 Subtract 3 from both sides. 2x = 12
Step 2. 2x/2 = 12/2 Divide both sides by 2. x = 6
...And that's where "move the +3 to the right and it becomes a -3" and "move the "2* to the right and it becomes a /2" come from. You are trying to "cancel" those terms on the left. Most math teachers and books don't really mention the reason behind this; they just drill it into the students via rote memorization. This can cause some issue later on when a student mistakenly performs the operations in the wrong order (divide first instead of subtract, which can easily happen if they are rote-memorize the usual order of operations), or when the student encounters a new operation like powers or roots or trigonometric functions or matrices or complex numbers and such. Instead of realizing the reason why these operations are performed, they start memorizing a complex web of order of operations and "take operation X to the right and it turns into operation Y" relations. And the dreaded "this one has multiple answers that I have to individually list out".
(The above can also be generalized to a vector of variables to solve for multiple unknowns simultaneously. Or you can solve and substitute for each variable one at a time.)
I think I always had trouble understanding because I never realized/was taught that = works both ways. Basically, I didn't understand the concept of equality. Usually those are written in exams or on the books like 2x + 3 = 15, so the whole "trick" of "send the non-x stuff to the other side" also meant to me that x should always be on the left side of the equation, = always had the meaning of an inference -> instead. I only understood how to do those things when I got bonked by analytical geometry when I was 16. Worst score of the class on the first exam, max score on the the last exam. Even got a box of chocolate bonbons as a reward for the hard work, yay.
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