Incredibly simple question about the logic behind systems of equations

[DrummingAtom]

I know this is a simple question but I can't exactly figure out the logic governing this problem, I just know it had to be this way. Let's say I have two equations:

x - 4y + 9 = 0
y - 3x + 5 = 0

If I set them equal to each other then I get no where because I'll have both x and y in the answer. But if I solve one of them for x or y then plug it into the other I can isolate one of them and then get an answer for x and y. Then by re-substitution I'll get answers for both x and y.

The only way setting them equal to each other would work is if x or y had values that canceled during that time which in this case they don't. Is there a mathematical logic rule governing this? Or is it just by practice we know it has to be done this way? Thanks

 

[Whovian]

I'm not sure I completely understand the question. Could you please elaborate a bit more?

 

[DrummingAtom]

I guess I'm asking if there is a logic statement of why this works that I described above. It seems that the combined equations of x - 4y + 9 = y - 3x + 5 is saying "AND" in logic but when they are separate they are "OR." So, only from the "OR" standpoint can the system be solved unless something cancels when in the "AND" form. Does that make sense?

I'm really having a hard time even thinking about what I'm looking for because this might be a dead end question.

 

[Whovian]

We already have that equation 1 AND equation 2 hold.

Anyways, we could multiply/divide both sides of one equation so stuff cancels.

 

[AlephZero]

If x - 4y + 9 = 0 you can multiply both sides the whole equation by any constant. For example multply by 2 and you get 2x - 8y + 18 = 0

So if you multiply by -3 to get -3x + 12y - 27 = 0, then you can "set them equal to each other" and say
-3x + 12y - 27 = y - 3x + 5 = 0
and the -3x cancels out.

If you want to solve the equations you choose to multiply by -3, and not by something else, to MAKE the x's cancel out

I'm not really sure what you are asking about, but does that explain what is going on?

 

[mathwonk]

setting them equal to each other is throwing away information. I.e. if both equations equal zero then of course they equal each other. why would you do that? You are trying to do something that takes advantage of both equations.

the whole idea is to notice that equations that look like this:

3 X + 7 Y = 6
2Y = 4,

are easy to solve, starting with the bottom one, then substituting the answer into the top one.

So the solution method is to try to replace two equations like yours, but two like these.

That is exactly what the substitution method does. I.e. solving for X in terms of Y, changes one of the equations into an equation that only has Y in it.

 

[MTannock]

I think you're asking if there's a logical rule that prevents you from finding x or y by making the two equations equal when they both equal zero.

And I don't think there is. To elaborate on what Whovian and AlephZero have said, the trick in this case is to multiply one of the equations by a number that will allow one of the variables to cancel out after you make the two equations equal. This works because both equations will always equal zero regardless of what you multiply them by, so you're not changing what they mean.

 

[arildno]

You CAN of course, proceed with the following two equations:
x - 4y + 9 = y - 3x + 5

y - 3x + 5 = 0

And then get the right answers.

Remember:
If you ONLY retain the equation x - 4y + 9 = y - 3x + 5 you have thrown away the information that one of the sides (and hence, both) actually equals zero.

 

[Whovian]

And please remember that the substitution method you present here is just another manifestation of substitution, which is also used to prove validity of adding equations and just plain old substitution.

 

[DrummingAtom]

You CAN of course, proceed with the following two equations:
x - 4y + 9 = y - 3x + 5

y - 3x + 5 = 0

And then get the right answers.

Remember:
If you ONLY retain the equation x - 4y + 9 = y - 3x + 5 you have thrown away the information that one of the sides (and hence, both) actually equals zero.

setting them equal to each other is throwing away information. I.e. if both equations equal zero then of course they equal each other. why would you do that? You are trying to do something that takes advantage of both equations.

Wow, thanks for all the responses.

Actually, these bring up the same point I think I was finding out when playing around with this: the throwing away of information part. I know it seems obvious but I guess I'm looking for a deeper reason as to why this works how it does. Going back to the "AND" and "OR" stuff I was talking about earlier. If one was given x - 4y + 9 = y - 3x + 5 that and ONLY that, x and y would be infinite solutions. Only with the information of that each side is equal to zero does a unique solution come about.

So, would it be correct in saying that an "AND" statement (the combined equation) has less information than an "OR" (the separate equations) statement? Or perhaps "AND" has less definite results?

 

[MTannock]

Not always, because as others have said, you can maintain the information if you make sure the multiple of one of the variables is the same in both equations before you make them equal.

 

[arildno]

You can look at this from the strictly logical point as well.
When we manipulate an equation properly, we SUBSTITUTE our original equation with one that is logically EQUIVALENT to it (contains the same information, AND the same solutions).
For example:
We have: 2x-5=0.
We can REPLACE that equation with the EQUIVALENT statement 2x=5

Now, when you COMBINE two equations, you are certainly doing something that is logically VALID, namely drawing an IMPLICATION that follows.

But, an implication is NOT the same as an equivalence; because some information was lost on the way.

We can go back from 2x=5 to 2x-5=0, but does x - 4y + 9 = y - 3x + 5 imply that y-3x+5=0 (couldn't y-3x+5=3.5 equally well??)That it MUST be zero is an additional information piece that is NOT contained in that equation you gained by combining your two original equations.

 

[Stephen Tashi]

So, would it be correct in saying that an "AND" statement (the combined equation) has less information than an "OR" (the separate equations) statement?

Your use of "AND" and "OR" in those descriptions doesn't make sense. The two simultaneous equations:
x - 4y + 9 = 0
y - 3x + 5 = 0
amount to the statement "x -4y + 9 = 0 AND y - 3x + 5 = 0".

The statement x - 4y + 9 = y - 3x + 5 is not an AND or OR of the statements involved.

The relevant logical pattern is this:

( (x - 4y + 9 = 0) and (y - 3x + 5 = 0) ) implies (x - 4y + 9 = y - 3x + 5).

From a statement of the form "A implies B", you can't conclude "B implies A". So from (x - 4y + 9 = y - 3x + 5), you can't conclude the original statements.

People who treat doing algebra as merely manipulating symbols fall into the trap of thinking that any permitted manipulation produces a step that can be reversed. The permitted manipulations in algebra are of the form:

(symbolic expression A) implies (you can write symbolic expression B)

Not all such steps are reversible because when you are given the symbolic expression B, you may not be able to coclude symbolic expression A.

For example:

if ( $x = \sqrt{25}$ ) then $x^2 = 25$.

Only x = 5 makes the first statement true, but both x = 5 and x = -5 make the second statement true.