Nice little equation: (4 - x)/(3 + x) = 16/25 Solve for x

[MisterGoof]

(4 - x)/(3 + x) = 16/25 Solve for x

I was told that I would need to use the distributive property when i cross multiplied, And i have no idea whatsoever how to do that..
Would anyone mind explaining how to do that? Or at least how to get started?

 

[wjm11]

Re: Nice little equation..

(4 - x)/(3 + x) = 16/25 Solve for x

I was told that I would need to use the distributive property when i cross multiplied, And i have no idea whatsoever how to do that..
Would anyone mind explaining how to do that? Or at least how to get started?

Cross multiply:

25(4 – x) = 16(3 + x)

Distributive property examples:

a(b + c) = ab + ac

a(b - c) = ab - ac

edited typo/error.

 

[Loren]

Re: Nice little equation..

You say "cross multiplied". I think you need to understand what is going on and what basic principle "cross multiplication" is a short-cut for. I'll give it a try this way.

(4 - x)/(3 + x) = 16/25 Solve for x

\(\displaystyle \frac{4-x}{3+x}=\frac{16}{25}\)

To eliminate working with fractions, multiply both sides of the equation by the least common denominator, which is 25(3+x).

\(\displaystyle \frac{4-x}{3+x}\cdot \frac{25(3+x)}{1}=\frac{16}{25}\cdot \frac{25(3+x)}{1}\)

Do the cancellation and you end up with the same thing as if you "cross multiplied".

25(4-x) = 16(3+x)

Now, simplify by using the distributive property, and solve for x. Be sure to check your answer. Why? Suppose you were to get x=-3. That would not be a viable result because it would involve division by zero, which is undefined.

 

[tkhunny]

Re: Nice little equation..

Whoever told you to "cross-multiply" should have his mouth washed out with soap. (My views. I welcome others'.)

One step at a time. Don't try to rush. Let the notation help you by WRITING OUT each step. Getting it on paper is WAY easier to organize than trying to keep it all in your head.

(4-x)/(3+x) = 16/25

Multiply both sides of the equation by (3+x), noting that 3+x better not be zero (0). Why is that?

[(4-x)/(3+x)]*(3+x) = (16/25)*(3+x)

Rearrange with appropaite properties of multiplication

[(4-x)*(3+x)]/(3+x) = [16*(3+x)]/25

There is an Associative property to use, now.

(4-x)*[(3+x)/(3+x)] = [16*(3+x)]/25

The answer to my thought question above applies to this step. We need an inverse property for multiplication.

(4-x)*[1] = [16*(3+x)]/25

Simplify

(4-x) = [16*(3+x)]/25

Multiply both sides by 25 (Why don't we have to worry about this being zero (0)?)

(4-x)*25 = {[16*(3+x)]/25}*25

Rearrange with appropaite properties of multiplication

(4-x)*25 = [16*(3+x)*25]/25

There is an Associative property to use, now.

(4-x)*25 = [16*(3+x)]*[25/25]

We need an inverse property for multiplication.

(4-x)*25 = [16*(3+x)]*[1]

Simplify

(4-x)*25 = 16*(3+x)

NOW we get to the Distributive Property of Multiplication over Addition. You'll remember it when you see it. Both sides.

4*25 - x*25 = 16*3 + 16*x

Simplify (Just multiplication)

100 - x*25 = 48 + 16*x

Can you take it from there?

Note: There is no need to jump to the end too soon. See how far we went before we used the "Distributive" hint?
Note: I didn't see any of that vulgar "c------m-----" going on in there.

 

[masters]

Re: Nice little equation..

Whoever told you to "cross-multiply" should have his mouth washed out with soap. (My views. I welcome others'.)

This treatise supports your argument, TK:

http://www.utdanacenter.org/mathtoolkit ... ssmult.pdf

And this one: http://mathforum.org/library/drmath/view/61108.html

breaks it down into its mathematical parts.

Unfortunately, many teachers teach the technique of "cross-multiplication" without explaining the mathematics behind it. That's why you see it come up so often in the forums.