11. The number of solutions to the following system of equations

y² – xy - |x|y + x|x| = 0

x² + y² = 1

is

The first equation factors:

Thus the values satisfying the first equation are the points (x, y) That fall on the lines y = x or y = |x|. Note that the line for y = x coincides with y = |x| for x >= 0. The lines y = x and y = |x| are different, however, when x < 0.

Consquently, the line y = x intersects the circle x² + y² = 1 at two points, one in the first quadrant and one on the third quadrant. The line y = |x| will have an additional point of intersection in the second quadrant