Please help, i'm in grade 9 and get exellent maths grades and everyone regard me smart but I keep asking myself why does this work and it gets really bothering Example when we studyed system of equations I start ask why does the substition method works, when we substitue the value will that make the equation (1) work, why does solving equation works.Then I start ask my self why do basic properties work why a/b * c=ac/b? and this is really confusing me please help what should I do I love mathsss Thanks everyone for answering I will tell you how I see for now why substitution (wich sometime is not convincing for me and I start looking for more details) so is this convincing: Consider the system: ax+by=c (1) a'x+b'x=c' (2) For x to be a solution of equation (1) it has to respect the relation x=(c-by)/a we substitute this value of x into the second equation for it to to be solution of both equation, we find y and then we find is this enough proof or I have to think deeper?
Please help I'm in grade 9 and excellent at maths but I keep asking why does it work
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I struggled with this far past grade 9, I would find myself trying to reprove basic basic facts in exams, because I forgot what the fact was or why it was true. Part of the problem is that in grade 9, you are not yet mature in recognizing the difference between what is definitional and what is a result. Another part of the issue is that you have not had enough time to internalize the basic facts so that they are second nature. The final part of the issue, is you are not yet comfortable knowing to what level of detail a concept deserves to be known at.
My answer is you should keep asking those questions to yourself, and try answering them. You will find the answers either in the moment, or they will become more natural as you mature mathematically.
For example, the fact $c\cdot \frac{a}{b} = \frac{ca}{b}$ is just a definition, or notation in grade 9. If you wanted a proof, you would cut a circular pizza into $b$ slices, take $a$ of them, then take $c$ of those. You can convince yourself that $c\cdot \frac{a}{b} = \frac{a}{b} + ... + \frac{a}{b}= \frac{ca}{b}$
Substitution is the idea that "for 2 equations to be satisfied at the same time, the unknowns must satisfy both equations at the same time". So you look for a relationship between the unknowns using eq(1), and you know that relationship must hold for eq(2) since the equations are satisfied for the same unknowns. Eventually you get some result like $x=5$, and work backwards through all the relations you found on the way.
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In mathematics it's important to distinguish between axioms and theorems. Axioms do not require a proof as we accept them as obvious facts. In case $\frac{a}{b}\cdot c$ we can use multiplication commutative law (which says that $x \cdot y=y \cdot x$) so $\frac{a}{b}\cdot c=c \cdot \frac{a}{b}$ and associative law: $c \cdot (a : b)=(c \cdot a) : b$
Both laws were considered as axioms for a long time until people like you started questioning them and came up with the proofs. You can refer to this question if you want to know more On the commutative property of multiplication (domain of integers, possibly reals)
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What you mention were mostly axioms, in that they are inherently self explanatory and thus does not require an extensive proof, unlike many of the theorems you see online and back in your 8th grade geometry.
I also went through that time questionong how stiff in math worled, leading me to being a better math student than I was before.
For one of your questions, the substitution property works via the rules of equality, nost of which are simple established axioms, in the sense that they need not any extensive proof to prove their logixal sense.
As you go higher in the realm of higher maths, you will begin to see even more strange results (see the recent 3b1b vid on colliding blocks) and studying the problem and the results can help find the underlying workings that makes the maths possible.
Have a great day!
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Thanks everyone for answering I will tell you how I see for now why substitution (wich sometime is not convincing for me and I start looking for more details) so is this convincing: Consider the system: ax+by=c (1) a'x+b'x=c' (2) For x to be a solution of equation (1) it has to respect the relation x=(c-by)/a we substitute this value of x into the second equation for it to to be solution of both equation, we find y and then we find is this enough proof or I have to think deeper? Also another question should I think in maths, because we have a we have b, or do a to have b like in substituion we get in a an equation a relation for x( to be solution of that equation or because it is solution to that equation) and should I think we replace the second equation by this relation ( to have the both equation have the same couple solution or because they have the same couple soulution (y=y,x=x) I am allowed to replace) Another example to make it clearer: Consider the following situation as an example clearer: Finding 3 numbers (lets call them a , b and c proportional respectivly to 2,5 and 6 and 3a+4b-2c=20 a,b,c prop. to 2 , 5 and 6 gives a/2=b/5=c/6=k which gives a=2k, b=5k ,6k then we replace them.... And here is the question should I think in my head that I replace a by 2k and b by 5k... for them TO BE prop to 2,5... or I am allowed to do that BECAUSE they are proportional to 2,5...?
The paradoxical thing is the basic and fundamental things are the hardest and we must learn them last and dedicate our lives to figuring them out. But if we learned them first we are doomed to never know anything.
Consider this analogy. If a child is hungry and asks what to do, you tell it to eat something. Why does this work? Why does eating satisfy hunger? Why does hunger need satisfying? Why does being hungry make us unhappy and why do we wish to avoid being unhappy? And what does eating have to do with it?
Well, ingesting and digesting food is a chemical process. There are molecular chains in foods called proteins and... I dont know. I never learned this stuff. I don't know ANYTHING about what I'm talking about. I don't know why eating food stops us from being hungry and keeps us alive. I don't know why staying alive is supposed to be a good thing or why I want to stay alive. None of this do I know.
But some people know and care very much and studied this. But they didn't study this when it first came up when they were two years old and feeling the pangs of hunger. At that time their parent said "You are hungry because you need food. Eat food and you won't be hungry". They didn't counter by saying "But I can't eat food unless I know why I need to eat food". If they worried about this they thought "I really want to know why" and when the got a chance they took high school biology and loved it and when they got to university they majored in biology because they really wanted to know the fundamentals.
The rest of us when asked "Why do we eat food when we are hungry" answer "What? Am I a biologist? 'Hungry' means you need food". And most of us when asked why does $\frac ab c = \frac {ac}b$ say "What? Am I a mathematician? Multiplication and division distribute. That's what they are." ... Unless we are mathematicians....
Point being: Fundamentals are hard. We spend years trying to properly describe them and figure them out.
Why does $\frac ab\times c = \frac {a\times c}b$. Well, that depends on our definitions and assumptions.
What is a number and what is multiplication and what is addition. Well, in the "real world" mass and volume is conserved so when you move amounts together and combine them or separate them into piles the amounts will consistently stay the same no matter what order we do them in.
Why? What am I? A physicist? I don't know. That's conservation of mass and the universe seems to like it... except in those weird quantum physics and ... why is there anything at all at all instead of nothing... and ... oh heck with it! I don't know. (There I said it!) That's how it is.
But I am a mathematician. The question I want to know is how can I describe these and break them into consistant logical definitions and properties.
Well, I define something as a "field". A field is a system of elements with properties. There are two "binary operations". A binary operation is a rule that if you take any two elements and you can combine them and you will always get an element in the system. If you combine the same two elements you will always get the same element and you can combine any two elements and you will always get an element.
Throwing rocks in a pile is a system that works like this. If our elements are the quantity of rocks and our "binary operation" is tossing one quantity of rocks together with a quantity of rocks, combined they get a quantity of rocks.
So we philosophize and observe and define that field is.
A collection of elements.
Two operations called "+" and "$\times$" even though we don't need to specify what those mean. They will mean different things in different systems. so that.
1) Addition and multiplication is commutative. It doesn't matter what order we do things in.
2) Addition and multiplication are associative. It doesn't matter how we group things as we decide what order we do things in.
3) Addition distributes over multiplication. That is if $a,b,c$ are three elements $a\times (b+c)$ will result in the same element that $a\times b + a\times c$.
Numerical quantities of rocks obey these rules because there's something funny in the way God made the universe. But, we mathematicians, with our brains were able to boil this down to the powerful three useful rules above which will serve our needs.
More rules.
4) For addition and multiplication: There is an element so that adding them to an element, or multiplying an elment by them the result is the element. In other words there is a "neutral" or "identity" element. For addition this number is "$0$" and $0 + a = a$. It is a rule that such an element exists. For multiplication this number is "$1$" and $1 \times a = a$.
5) For any element there is a number that "undoes" it so that can get back to the identity. This is called an "inverse". The additive inverse is such that if we have $a$ then there is some element $a'$ so that $a + a' = 0$. We notate this as $-a$ so that by definition $-a$ is the element so that $a+(-a) = 0$. For multiplication we notate this either as $\frac 1a$ or as $a^{-1}$. By definition $\frac 1a$ is the element so that $a \times \frac 1a = 1$.
But there is one catch. For every $a$ EXCEPT $0$ there is a $\frac 1a$ so that $a\times \frac 1a = 1$, but there is no such number for $0$. This would give an inconsistent result and our system would fail.
(Let $k = a\times 0$ then $0 = k + (-k)$. $k = a\times 0= a\times (0 + 0) = a\times 0 + a\times 0 = k + k$ So $0 = k + (-k) = k + k + (-k) = k + (k+(-k)) = k + 0 = k$ os $k = a\times 0 = 0$. So if $\frac 10$ exist we'd have $0 = 0 \times \frac 10 = 1$ so $0=1$. Which would mean $a = a\times 1 = a\times 0 = 0$ so we only have one element. The element $0$. This is actually consistant but not useful. So if we want a "field" with more than one element we have a rule:)
6) $0 \ne 1$ and there is no $\frac 10$ so that $0\times \frac 10 = 1$.
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Given all this.
Because multiplication is a binary operation we know that for any $b$ and any $a\ne 0$ that an element $a \times \frac 1b = \frac 1b \times a$ exists. We can notate this element as $\frac ab$.
So the statement $\frac ab \times c = \frac {ac}b$ means:
The number $(a\times k)\times c$ where $k$ is the number so that $b\times k = 1$ is equal to $a\times (k \times c)$ (because of associativity) which is equal to $a\times (c \times k)$ (by commutativity) = (a\times c)\times k$ (associativity again) = \frac {ac}b$ (just notation).
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In other words. Questioning is good. But the answer is that we have to lay fundamental groundwork and axioms and identify precisely what they are first.