James paddles upstream in a canoe at 2mph (relative to the shore), and when he paddles down-stream, he goes 9mph. Find the speed of the current and the speed James can paddle in still water and come up with an equation.
Not sure what equation you're looking for, but against the current he goes 2 mph and with he goes 9mph, so 9-2=7mph difference between them, divided by 2 = 3.5 mph of the current. So in this case, James can paddle 2+3.5= 5.5 mph in still water and the current moves at 3.5 mph.
James can paddle 5.5 mph
The current moves at 3.5 mph
Equation: speed = 5.5 + 3.5x where x can equal -1, 0, or 1 depending on whether James is going with the current (x = 1), without the current (x = 0), or against the current (x = -1)
Let x represent James's speed and y represent the speed of the current. Since upstream the current goes against him, \(x-y=2\) describes the speed upstream, and since downstream the current goes with him, \(x+y=9\) describes the speed downstream.
Use elimination to solve for x: \(x-y=2 \\ -(x+y=9) \\ =-2y=-7 \\ y=7/2\) or 3.5 miles per hour for James in still water. Substitute y into either original equation: \(x-(7/2)=2 \\ x=11/2\) or 5.5 miles per hour for the current. As far as an equation, you can use \(y=(11/2)x\), where x is the number of hours James has paddled in still water and y is the distance he has traveled (in miles).