Find $x+y$ for the given conditions

Question

Here's a question that was asked in the International Kangaroo Math Contest 2017. Since I get a little confused when I solve equations having the absolute function, so I couldn't get the required answer.

The Question:

What is the value of $x+y$ if $\vert x\vert+x+y=5$ and $x+\vert y\vert-y=10?$

The Answer: The answer given in the key is $x+y=1$.

What I Tried:

I first added both the equations to get $$\vert x\vert+\vert y\vert+ 2x=15$$ So I got nothing. Then I subtracted the second equation from the first to get $$\vert x\vert -\vert y\vert+2y=-5$$ And still I got nothing. I can't figure out any other operation that I can perform on these equations. So please help me with this.

Thanks for the attention.

Answer 1

You need only check the two cases

• $y \geq 0 \Rightarrow x+|y|-y =x = 10 \stackrel{1st\:eqn.}{\Rightarrow}y < 0 \mbox{ Contradiction!}$
• $x \leq 0 \Rightarrow |x|+x+y = y =5 \stackrel{2nd\:eqn.}{\Rightarrow}x=10 \mbox{ Contradiction!}$

So, if there is any solution, then $x>0$ and $y < 0$. Now you need only solve \begin{eqnarray*} 2x + y& = & 5\\ x - 2y & = & 10 \end{eqnarray*} $\Rightarrow x= 4$ and $y =-3$.