By linearity of the cross product, \[(5 a + 7 b) \times (-a + 3 b) = -5 a \times a + 15 a \times b - 7 b \times a + 21 b \times b.\]We have that $ a \times a = b \times b = 0 $ and $ b \times a = -a \times b $, so \[-5 a \times a + 15 a \times b - 7 b \times a + 21 b \times b = 0 + 15 a \times b + 7 a \times b + 0 = 22 a \times b.\]The answer is $ k = \boxed{22} $.