Question 60579
a) To see if there is a possibility of a collision, you could graph these two linear equations that represent the paths of the ships.
First, get the first equation into the slope-intercept form:
2x+3y = 6 Solve this for y by subtracting 2x.
3y = -2x+6 Now divide both sides by 3.
y = (-2/3)x + 2
Now the graph of the two equations looks like:
{{{graph(300,200,-5,5,-5,5,(-2/3)x+2,(2/3)x-3)}}}
As you can see, there is a possibility of a collision if the two ships continue along their respective paths.

b) The danger point is the intersection of the two lines and this can be found by solving this system of equations:
y = (-2/3)x + 2
y = (2/3)x - 3
Set these two equations equal to each other and solve for x.
(2/3)x-3 = (-2/3)x+2 Simplify and solve for x.
(4/3)x = 5
x = 15/4 Now substitute this into either equation and solve for y.

y = (2/3)(15/4)-3
y = (5/2) - 3
y = -1/2

The coordinates of the danger (collision) point are: (15/4, -1/2)

c) Is a collision a certainty? 
No, it is not.  Whether the two ships collide or not depends on their individual speeds. A collission can be avoided by appropriate adjustment of their speeds even if they do remain on the same paths as indicated on the graph.