The surface area of the pyramid is the area of the square plus the area of each of the triangle faces. The area of a square is length squared. We know the length is 5, so the area Asquare = 52 = 25.
The area of a triangle is 1/2 base times height. In this case, the base is 5 and the height is the slant height, h (to visualize this, look at a picture of pyramid slant height). The equation is: Atriangle = 1/2 (5) (h) = 2.5h.
There are four triangles, so the total surface area is Asurface = Asquare + 4 (Atriangle) = 25 + 4 (2.5h) = 25 + 10h. We know the surface area is 175, so:
25 + 10h = 175
10h = 150
h = 15
The slant height is 15 inches.