Given the lines:

y= 3(x-2)

2y = 6x -9

We need to determine if the lines are parallel.

First we will rewrite he equations of the lines into the slope form.

If the slopes are equal, then the lines are parallel.

==> y= 3(x-2)

==> y= 3x -6 ==> the slope m1 = 3

2y = 6x -9

Divide by 2:

==> y= 3x - 4.5 ==> the slope m2= 3

Then, we notice that m1= m2.

**Then, the lines are parallel.**

Two parallel do not intersect.

Now, the lines we have are:

y= 3(x-2) and 2y = 6x -9

2y = 6x -9

=> y = ( 6x - 9)/2

=> y = 3x - 9/2

Now we see that y= 3(x-2) => y - 3x = -6 and y = 3x - 9/2 => y - 3x = -9/2 are two non - intersecting lines.

**Therefore they are parallel.**

We'll verify if the system formed from the equtaions of the given lines has solutions. If so, that means that the lines are intercepting. If the system has no solutions, the line are parallel.

We'll substitute the expression of y (the 1st line equation) into the 2nd equation.

6(x-2) = 6x -9

We'll remove the brackets:

6x - 12 = 6x - 9

We'll eliminate like terms;

-12 = -9, not true.

**So, the system has no solution, meaning that the lines are parallel.**