I do not know how you obtained the above, since on V 10.02, it does not solve it as shown
Clear[y1, x1, y2, x2, a, b];
eq1 = y1 == a*x1^b;
eq2 = y2 == a*x2^b;
Solve[{eq1, eq2}, {a, b}]
gives
Solve::nsmet: "This system cannot be solved with the methods available to Solve.
But you can get the solutions use this:
Clear[y1, x1, y2, x2, a, b];
eq1 = y1 == a*x1^b;
eq2 = y2 == a*x2^b;
Solve[{eq1, eq2}, {a, b}, MaxExtraConditions -> All]