The equation of the line L1 is y = 3x – 2. The equation of the line L2 is 3y – 9x + 5 = 0. Show that these two lines are parallel.

The equation of a straight line is usually given in the form y = mx + c. To prove that two lines are parallel we will need to show that both lines have the same gradient. The gradient of a line is the slope/steepness of the line and it is represented by 'm' (the number that comes before the 'x') when the equation is written in the form y = mx + c. The line L1 is already given in the form y = mx + c, so if we rearrange the equation of the line L2 into the same form we can prove the two lines are parallel by showing both have the same gradient (the same number as 'm'). When rearranging 3y - 9x +5 = 0 we want to have 'y' part on one side and everything else on the other. To do this we first need to subtract 5 from both sides of the equation. Remember - whatever we do to one side of the equation must be done to the other side too. By subtracting 5 from both sides we will be left with 3y - 9x = - 5. We then need to add '9x' to both sides in order to get the 'y' part on its own. By doing this we are left with 3y = 9x - 5. We now have the 'y' part on one side of the equation, but our equation has '3y' when we just want 'y' to allow us to show the equation in the form y = mx + c. To get from '3y' to 'y' we must divide by 3. Remember - if we divide '3y' by 3 we must also divided '9x' and '- 5' by 3 too. By dividing everything by 3 we are left with y = 3x - (5/3). We have shown the line L2 can also be written as y = 3x - (5/3). We can now see the gradient of the line L2 is 3. By looking at the equation of L1 (already given in the form y = mx +c) we can see the line L1 also has a gradient of 3. This proves that the lines L1 and L2 are parallel, because they both have the same gradient.