Solving a set of equations with absolute values

Question

How would you go about solving equations with absolute values? ~

$$ |x − 1| + |y − 5| = 1 $$ $$ |x − 1| − y = −5 $$

Answer 1

from the second equation we get $$y+|y-5|=6$$ if $$y\geq 5$$ we have $$2y-5=0$$ thus $$y=\frac{11}{2}$$ if $$y<5$$ we have $$y-y+5=6$$ there is no solution in this case. computing $x$ we get the following solution set $$x=\frac{3}{2},y=\frac{11}{2}$$ or $$x=\frac{1}{2},y=\frac{11}{2}$$

Answer 2

Hint:

Apply this rule: Suppose $f(x)=|x-k|$, then $f(x)$ can be decomposed into: $$ f(x)= \begin{cases} x-k, & \text{when } x>k;\\ 0, & \text{when }x=k;\\ -(x-k), & \text{when }x<k. \end{cases} $$