Final answer:
To find the point at which the lines intersect, we need to set the two lines equal to each other, solve for the values of r, s, and t, and substitute those values back into one of the line equations. The intersection point will be the values that satisfy both equations.
Step-by-step explanation:
To find the point at which the lines intersect, we first need to set the two lines equal to each other and solve for the values of r, s, and t. The given lines are r = 3t + 2 and r = 5s - 2. Once we have the values of t and s, we can substitute them back into one of the line equations to find the corresponding values of r. The intersection point will be the values of r, s, and t that satisfy both equations.
Setting the two lines equal to each other:
3t + 2 = 5s - 2
Rearrange the equation to solve for s:
5s = 3t + 4
s = (3t + 4)/5
Substitute this value of s into one of the line equations:
r = 5((3t + 4)/5) - 2
r = 3t + 4 - 2
r = 3t + 2
So, the lines intersect at the point (r, s, t) = (3t + 2, (3t + 4)/5, t).
We can substitute any value of t into this expression to find the coordinates of the intersection point.