The line AB has equation 5x + 3y + 3 = 0 and it intersects the line with equation 3x - 2y + 17 = 0 at the point B. Find the coordinates of B.

Two straight lines can intersect at only one point. To identify the point at which the lines cross, we make a simultaneous equation, and solve x and y from here, by firstly writing one equation on top of the other:

1. 5x + 3y + 3 = 0

2. 3x - 2y + 17 = 0

We then need to make the equations so that the number preceding either the x or the y is the same in both equations. The simplest way to do this here is to mutliply the top line by 2 and the bottom line by 3, in order to make both equations have 6y in them.

1. 10x + 6y + 6 = 0

2. 9x - 6y + 51 = 0

From here, we can remove the 6y from both lines and add the equations to solve for x as one is negative and one is positive (if they were both the same sign, we would have to remove one from the other to solve for x).

19x = -57 therefore x = -3

Then we need to find the y coordinate so we sub in x to one of the equations:

1. -15 + 3y + 3 = 0 so 3y = 12 so y = 4

To check this we sub y and x into our other equation:

-9 - 8 + 17 = 0, thus the coordinates are (-3,4).