Solving Homework Statement: Proving a/b=c/d=e/f to ace/bdf

[Miike012]

Homework Statement

if a/b = c/d = e/f

then show that...

(a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3) = ace/bdf

Homework Equations

1.I understand how to prove it but I don't understand algebraically how they got from
a/b = c/d = e/f to... (a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3)
( or is this irrelevant and what I should focus on is the proving the theorem?)

2. second I don't understnad how (a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3) = ace/bdf?
the numerator does not simplify into the denominator UNLESS...
because a/b = c/d = e/f

(a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3) could also equal (k^4 + 2k^3 - 3k^4)/(k^4 + 2k^3 - 3k^4) = 1

and ace/bdf could equal k^3/k^3 = 1

therefore 1=1 is true?

 

[tiny-tim]

Hi Miike012!

Start again …

"if a/b = c/d = e/f = p … "

 

[Miike012]

then
a=bp, c= dp, and e=fp


I understand that part... is there something that you are hinting at?

 

[vela]

1.I understand how to prove it but I don't understand algebraically how they got from
a/b = c/d = e/f to... (a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3)
( or is this irrelevant and what I should focus on is the proving the theorem?)

It's irrelevant. The expression is a given. You don't care where it comes from.

2. second I don't understnad how (a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3) = ace/bdf?
the numerator does not simplify into the denominator UNLESS...
because a/b = c/d = e/f

The point of the problem is to show that if you assume a/b = c/d = e/f, then (a³b+2c²e-3ae²f)/(b⁴+2d²f-3bf³) = ace/bdf. All proofs are like this. You're given a set of assumptions, and you want to show that it then follows that some statement is true.

(a^3b + 2c^2e - 3ae^2f) / (b^4 + 2d^2f - 3bf^3) could also equal (k^4 + 2k^3 - 3k^4)/(k^4 + 2k^3 - 3k^4) = 1

and ace/bdf could equal k^3/k^3 = 1

therefore 1=1 is true?

It's certainly true if a, b, c, d, e, and f all equaled each other (and aren't equal to 0), the equality would hold, but that fact doesn't really help in your proof because you're using an assumption that wasn't a given in the problem. All you can assume is a/b = c/d = e/f.

 

[Miike012]

wow, thank you ... that makes a lot more sense now.

 

[vela]

A good way to write a proof like this is to start with one side of the equation and manipulate it until you end up with the other side:

$$\begin{align*} \frac{a^3b + 2c^2e - 3ae^2f}{b^4 + 2d^2f - 3bf^3} & = \cdots \\ & = \cdots \\ & \vdots \\ & = \frac{abc}{def} \end{align*}$$