Understanding Proofs and Problem Solving Techniques | Help and Explanation

[Miike012]

Homework Statement

***I already proved the problem and came up with a solution. My true issue are the questions below.***
If : a/b = c/d = e/f ; Let equal to k then: k -----> a = kb ; c = kd ; e = kf

Prove : (2a^4b^2 + 3a^2e^2 - 5e^4f) / (2b^6 + 3b^2f^2 -5k^4f^5) = a^4/b^4

The Attempt at a Solution

After imputing values I get:

k^4(2b^6 + 3b^2f^2 -5k^4f^5) / (2b^6 + 3b^2f^2 -5k^4f^5) = k^4b^4/b^4

After simplifying I get:
k^4 = k^4

(c) Question:
1. Is the reason why we set a/b = c/d = e/f equal to "k" is because I am supposed to assume that if a/b = c/d = e/f, then I can set them all equal to a single variable because after all they are equal to one another? If I am incorrect will some one explain it to me a little better?
2. When trying to prove problems in general... is the main goal to get the equation equal to zero? After all if k^4 = k^4 then if I subtract k^4 on one side then 0 = 0.
3. If there is anything else that you think will be valuable for me to understand when proving theorms such as this please let me know.

Thank you.

 

[rock.freak667]

(c) Question:
1. Is the reason why we set a/b = c/d = e/f equal to "k" is because I am supposed to assume that if a/b = c/d = e/f, then I can set them all equal to a single variable because after all they are equal to one another? If I am incorrect will some one explain it to me a little better?

Yes that is essentially why they are all equal to some constant k.

2. When trying to prove problems in general... is the main goal to get the equation equal to zero? After all if k^4 = k^4 then if I subtract k^4 on one side then 0 = 0.

Well in this case, you chose the left side in an attempt to to make it the same as the right side.

your RHS simplified to k⁴ and since a/b = k from the initial proposition, it follows that your RHS is a⁴/b⁴

 

[vela]

To elaborate a bit on what rock.freak meant in his last sentence: It's better to work with just one side of the equation and try to turn it into the other side, rather than messing with both sides and showing the two end up being the same thing as you did. In other words, write something like

$$\begin{align*} \frac{2a^4b^2 + 3a^2e^2 - 5e^4f}{2b^6 + 3b^2f^2 -5f^5} & = \frac{2(kb)^4b^2 + 3(kb)^2(kf)^2 - 5(kf)^4f}{2b^6 + 3b^2f^2 -5f^5} \\ & \vdots \\ & = k^4 \\ & = \left(\frac{a}{b}\right)^4 \\ & = \frac{a^4}{b^4} \end{align*}$$

Then from the transitive property of equality, you can conclude that the left-hand side equals the right-hand side.

One problem with writing

$$\begin{align*} \frac{2a^4b^2 + 3a^2e^2 - 5e^4f}{2b^6 + 3b^2f^2 -5f^5} & = \frac{a^4}{b^4} \\ & \vdots \\ k^4 & = k^4 \end{align*}$$

is that in writing the first line, you're assuming what you're trying to prove. Also, just because that assumption leads to a statement that's true, logically, you still can't conclude that the assumption was valid.

 

[Miike012]

I pic one side and try to set it to the other side? But then you said just use the transitive property? If I use the trans. prop I wouldn't have to pick one side and try to turn it into the other side...

 

[vela]

If you write A=B=C=D=...=Z, that's shorthand for A=B, B=C, C=D, ..., Y=Z, and the transitive property of equality let's you say A=Z.

 

[Miike012]

Yes I understand that property. Is that what you mean when you said "It's better to work with just one side of the equation and try to turn it into the other side."

 

[vela]

Yes.

 

[Miike012]

Sorry, misunderstanding on my end. I thought you ment pic one side of k^4(2b^6 + 3b^2f^2 -5k^4f^5) / (2b^6 + 3b^2f^2 -5k^4f^5) = k^4b^4/b^4... either k^4(2b^6 + 3b^2f^2 -5k^4f^5) / (2b^6 + 3b^2f^2 -5k^4f^5) OR k^4b^4/b^4. Then use algebra to try to make the right side look like the left or viseverse.

 

[vela]

That is exactly what I meant.

 

[Miike012]

nevermind thast wouldn't work, because I am assuming that it equals 0.

 

[vela]

No, there's no setting anything to 0 involved. You're making this way too complicated.

 

[Miike012]

I just don't understand how you can solve for only one side and neglect the other side. Because I thought if you add,subtract,mult or divide.. you have to do it to the other side.

 

[vela]

You're not solving anything, and you don't have an equation to start with. You're trying to show that two expressions are equal.

 

[Miike012]

Ok for example say after substitution I come up with

3a = 12b
If I pic the right side I would multiply by 4b/a which gives me 12b this 12b = 12b? is that correct... if that's so that totally doesn't make sense because I could easily make the right look like the left or vise vera if I can mult or divide or subtract or add anything I want. There has to be some correlation connecting the two.

 

[Mentallic]

No no, that's not the same. In your original question you don't multiply by anything, you instead use algebraic techniques such as factorizing, and given the info that a/b = ... = k you can simplify that long expression down to just a^4/b^4.
It's the same as saying: simplify 10²/5² given that 10/5=2 in a sense. Once you simplify it down to 4, you haven't changed anything.