If runners in a long distance race were to run straight from the starting line to the finish line they would run 13 kilometers. however, the road they run makes them travel longer than that. they must run 5 kilometers south and then head west "x" kilometers for the remainder of the race. how far do the runners travel?

This "road" they are running is basically an L-shape, and combined with the direct distance from start to end forms a triangle. This triangle has a hypotenuse of 13 (since the direction of the road changes from south to west, we can pretty much assume that it is a right triangle with legs of 5 and X.)
 So to find X, we use the Pythagorean Theorem.
a²+b²=c²

(5)²+X²=13²
25+X²=169
X²=144
Distance cannot be negative, so we know that X=12. This is the length of the remainder of the road.
To find total distance, just add the two lengths of road together.
5+12=17