Checking if operation is associative

Question

I am given the following exercice.

Is the operation $x ∆ y = 2xy - 3x + 3y$ Associative on the set of real number?

In order to do this, one must check that

$(x ∆ y) ∆ z = x ∆ (y ∆ z)$

So answer provided is:

$( x ∆ y ) ∆ z =(2xy - 3x + 3y) ∆ z = 4xyz - 6xz + 6yz - 6xy + 9x - 9y + 3z$

$x ∆ (y ∆ z) = x ∆ (2xy - 3x + 3z) = 4xyz - 6xy + 6xz - 3x + 6yz - 9y + 9z$

This shows that both sides aren't the same, thus the operation is not associative.

However, I have no idea how this is done. I don't understand where to value of z if being defined or how it is calculated.

Any help would help, I'm very stuck. Thanks

Answer 1

First "equation": $(x \Delta y) \Delta z$

First we have to calculate $(xΔy)$ i.e. $(2xy−3x+3y)$ : call it $k$.

Then we have to calculate $(kΔz)$ i.e. $(2kz−3k+3z)$.

Finally, we have to "plug in" $(2xy−3x+3y)$ in place of $k$ in the previous expression to get :

$2(2xy−3x+3y)z-3(2xy−3x+3y)+3z$.

This in turn is :

$4xyz-6xz+6yz-6xy+9x-9y+3z$.

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Second "equation": $x \Delta (y \Delta z)$.

First calculate $(yΔz)$ i.e. $(2yz−3y+3z)$ : call it $k$.

Then calculate $(xΔk)$ and complete it in the same way.