Algebraic Reasoning Proofs and Properties...help?

Question

Can I use a property backwards? Such as if (k+k) +2 = (1 x k + 1 x k) +2 , would that be the identity property of multiplication, even if it's backwards? Also, for 2k+2x1 = 2(k+1), is that the distributive property? Or would it just be "factoring"? What would I write in the reasons column of the proof? Basically, I'm just wondering if these properties worked backwards. If not, could you tell me what property it is called?

Also, if it's like:

3x + (-12) = 0,

[3x + (-12)] +12 = 0+12

What property is that?

Thanks for your help :)

P.S. Please don't make rude remarks such as "DO YOUR HOMEWORK YOURSELF!" or something, because I AM TRYING. The school library gave me the wrong textbooks for my algebra 2 class, so basically I have nothing to look at to figure this out by myself. So it'd help both me and you if you didn't say something rude.

=)

Answer 1

Some properties work backwards, and others don't.

Properties that assert two things are equal (like 1 x k = k) will always work backwards. The property "1 x k = k" says that "k" and "1 x k" really mean the same thing, so we can use them interchangeably--we can replace "k" with "1 x k", or we can replace "1 x k" with "k".

You can replace "k" with "1 x k" and write "identity property of multiplication" in the reasons column. That's fine.

---

The distributive property has an = in it (it asserts that a(b + c) = ab + ac ) so it also works backwards.

You can replace "2k + 2 x 1" with "2(k + 1)" and write "distributive property" in the reasons column. That's fine.

---

The last property you listed is called the additive property of equality. It says that if two things are equal, then when you add the same thing to them both, they'll still be equal.