Another x, Another y

To check in which quadrant the solution lies, I sketch the two curves (attached Figure)

Now $|x|+x+y=8$ is $2x+y=8$ for $x\geq{0}$ and $y=8$ for $x\leq{0}$ ... the discussion of the other equation is analogous. We see that the solution lies in the fourth quadrant, where $x>0,y<0$ , with $|x|=x$ and $|y|=-y$ , so that we have to solve the equations $2x+y=8, x-2y=14$ . Substituting $y=8-2x$ , we find $x=6,y=-4$ . Thus $x+y=\boxed{2}$

1. Confirm $x$ is not a negative number and thus proving $|x| = x$ (since |x| = x for non-negative numbers)

Assume $x$ is negative number $a$ , that is $-a$

$|x| + x + y = 8$

$|-a| + (-a) + y =8$

$a - a + y = 8$

$y = 8$ that means y is a positive number

if $y = 8$ , then

$x + |y| - y = 14$

$x + |8| - 8 = 14$

$x = 14$ since x became positive, it disagreed with our assumption that x is a negative number, so therefore x in not a negative number and thus prove $|x| = x$

2. Confirm $y$ is a negative number, proving $|y| = -y$ (since y will be negative to begin with, -y is positive)

let $y$ be a positive number

$x + |y| - y = 14$

$x + y - y = 14$

$x$ = 14

if $x$ = 14, then

$|x| + x + y = 8$

$|14| + 14 + y = 8$

$28 + y = 8$

$y = -20$ , y became negative which negates our assumption that y is positive so therefore it is not positive, proving $|y| = -y$

3. Find y. Since|x| = x, substitute x for |x|

$|x| + x + y = 8$

$2x + y = 8$

$y = 8 - 2x$

4. Solve for x. Since|y| = -y, substitute -y for |y|

$x + |y| - y = 14$

$x - 2y = 14$ , y = 8 - 2x

$x -2(8 - 2x) = 14$

$x - 16 + 4x = 14$

$5x = 30$

$x = 6$

5. Solve for y.

$y = 8 - 2x$

$y = 8 - 2(6)$

$y = 8 - 12$

$y = -4$

6. Solve x + y

$6 -4 = \boxed{2}$

We can consider equations with absolute functions by inverting the $\pm$ signs of variables $\{x,y\}$ at $|f(x,y) < 0|$ . It is equivalent to drawing the graphs.

$\begin{cases} |x|+x+y=8 & \Rightarrow \begin{cases} 2x + y = 8 & \text{for } x \ge 0 & ...(1a) \\ y = 8 & \text{for } x < 0 & ...(1b) \end{cases} \\ x+|y|-y=14 & \Rightarrow \begin{cases} x = 14 & \text{for } y \ge 0 & ...(2a) \\ x-2y = 14 & \text{for } y < 0 & ...(2b) \end{cases} \end{cases}$

Consider all the cases:

$\begin{cases} \text{Eq. 1a \& Eq. 2a} & \Rightarrow 28+y = 8 & \Rightarrow y = -20 \color{#D61F06}{< 0} & \color{#D61F06}{\text{Rejected}} \\ \text{Eq. 1b \& Eq. 2a} & \Rightarrow y = 8 & \text{but } x = 14 \color{#D61F06}{> 0} & \color{#D61F06}{\text{Rejected}} \\ \text{Eq. 1b \& Eq. 2b} & \Rightarrow y = 8 \color{#D61F06}{> 0} & & \color{#D61F06}{\text{Rejected}} \\ 2\times \text{Eq. 1a} + \text{Eq. 2b} & \Rightarrow 5x = 30 & \Rightarrow x = 6 \color{#3D99F6}{> 0} \\ & \Rightarrow 12 + y = 8 & \Rightarrow y = -4 \color{#3D99F6}{< 0} & \color{#3D99F6}{\text{Accepted}} \end{cases}$

$\Rightarrow x + y = 6-4 = \boxed{2}$

One solution would be to determine the nature of x and y first: From the first equation, ∣x∣+x+y=8, isolate the absolute value of x: ∣x∣ = 8 - x - y Because it is unclear if x is positive or negative, split the equation into cases. If x is positive, then x = 8 - x - y, because ∣x∣ = x if x > 0. Simplifying this equation gets 2x + y = 8. If x is negative, then x = -8 + x + y, because ∣x∣ = -x if x < 0. Simplifying this equation gets y = 8. One of the above equations must be false. y = 8 is clearly false because it would mean that x = 0, which does not agree with x + |y| - y = 14. Therefore, 2x + y = 8 is true and x > 0. The same technique can be applied to the second equation of x + |y| - y = 14, and isolating |y|. If y is positive, then x = 14. If y is negative, then -x + 2y = -14. x = 14 is false because it means that y = 0, which does not agree with ∣x∣ = 8 - x - y. So -x + 2y = -14 is true and y < 0. By setting up a system of equations with 2x + y = 8 and -x + 2y = -14, x = 6 and y = -4, so x + y = 2.

There are four possible cases: (x>0,y>0),(x<0,y<0),(x>0,y<0),(x<0,y>0). After testing, x>0, y<0 seems to work. As x is already positive, the first equation would be 2x + y = 8. The since y is absolute, y becomes -y. -y-y = -2y. Therefore the second equation is x - 2y = 14 Solving, we get x = 6, y = -4. x + y = 2.

X=6,y=(-2) so, x+y=2