Acar traveling north at 10.0 m/s crashes into a car traveling east at 15 m/s at an unexpectedly icy intersection. the cars lock together as they skid on the ice. the two cars have the same mass. what is their combined speed after the collision?

The first thing you should know is the conservation of the linear momentum
 Pi = Pf
 We have then that before the shock:
 Pi = mvi1 + mvi2 = m (10.0) j + m (15.0) i
 We have after the shock:
 Pf = 2mvf = 2mv (sinx) j + 2mv (cosx) i
 Matching both expressions:
 m (10.0) j + m (15.0) i = 2mv (sinx) j + 2mv (cosx) i
 Rewriting
 (10.0) j + (15.0) i = 2v (sinx) j + 2v (cosx) i
 We have 2 equations (components j and i) and two unknowns (angle x and v)
 2v (senx) = 10.0
 2v (cosx) = 15.0
 Resolving:
 tanx = (10.0 / 15.0)
 x = atan (10.0 / 15.0) = 33.69 degrees
 Clearing v
 2v (senx) = 10.0
 v = (10.0 / 2) * (1 / sen (33.69)) = 9.01 m / s
 answer
 their combined speed after the collision is 9.01 m / s in the direction 33.69 degrees