Justify each step with a property, definition or an operation

Question

A math expression has changed from one form to another and I need to know what has happened during that step:

1. from (3x + 10x) + 2y to x(3 + 10) + 2y

we have two different answers: one says it is a result of a distributive operation and another says it is a result of an associative operation.

which one is correct, distributive or associative?

2. from x(13) + 2y to 13x + 2y

which one is correct, distributive or commutative?

thanks,

Answer 1

$(3x+10x)=(x3+x10)$ by Commutative and $(x3+x10)=x(3+10)$ by distributive.

But also: $(3x+10x)=(3+10)x$ by distributive and $(3+10)x=x(3+10)$ by commutative

The same is for the $x(13)$(In this case we may include that $a+0=a$ and that $a\cdot0=0$ for all $a$)

Associative is only in the case of $f(f(a,b),c)=f(a,f(b,c))$.

For example saying that $(a+b)+c=a+(b+c)$ is by associative: $f(a,b)=a+b$ so $f(f(a,b),c)=(a+b)+c$ that is equal to $a+(b+c)=f(a,f(b,c))$.

Answer 2

On the first one you can factor an x so yes, you can say call out the distributive property. On the second question you are commuting the first term. Remember that multiplication is commutative.

Answer 3

Distribution is the property that $a(b+c) = ab + ac$? (i.e. you are distributing multiplication across a sum). Distribution goes the other way in that $ab + ac = a(b+c)$. (i.e. you can factor terms out that have been distributed.)

Association of addition is the property that $(a+b) +c = a+(b+c)$ (i.e. it doesn't matter how you group them)

Association of multiplication is the property that $a(bc) = (ab)c$ (ditto).

Which does $(3x + 10x) + 2y \implies x(3+10) + 2y$ appear to be. Clearly distributive.

Perhaps there was a term before that $3x + (10x + 2y) \implies (3x + 10x) + 2y$ is associative.

$x(3+10) + 27 \implies x(13) + 27$ is just addition.

And $x(13) + 27= 13x + 27$ is just reversing the order of $13$ and $x$. That we can reverse order of multiplication is the commutative property of multiplication.