The lines y = 3(x-2) and 2y = 6x -9 are parallel if they have the same slope.
Express the equation of the lines in the form y = mx + c
y = 3x - 6 and y = 3x - (9/2)
The slope in both cases is 3.
This proves that the lines 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.