Systems of linear equation with absolute value????????????????????

Question

solve the system

3lx-yl - 2lx+yl = -35

4lx+yl - 17lx-yl = 15

i need solutions :) thanks

Answer 1

Let u = |x - y| and v = |x + y|.  The system becomes

3u - 2v = -35

-17u + 4v = 15.  The solution is u = 5 and v = 25.  Now we consider 4 cases:

Case I.  x - y >= 0 and x + y >= 0.

In this case |x - y| = x - y and |x + y| = x + y.  The system is

x - y = 5

x + y = 25.  We find x = 15 and y = 10.

Case II.  x - y < 0 and x + y >- 0.

Now |x - y| = y - x and |x + y| = x + y.  The system is

y - x = 5 and x + y = 25.  The solution is y = 15, x = 10.

Case III.  x - y >= 0 and x + y < 0.

The system is x - y = 5 and -x - y = 25.  Solution is y = -15, x = -10.

Case IV.  x - y < 0 and x + y < 0.

The system is -x + y = 5 and -x - y = 25, with solution x = -15, y = -10.