Area of ABS Region

It is hard to work with this equation because there are weird terms inside the absolute values. However, we recognize the $ax+by$ and $bx-ay$ pattern, which suggests rotating this graph to get something easier.

So, let's apply the rotation $\left\{\begin{array}{l}x'=x+2y\\ y'=2x-y \end{array}\right.$

To figure out exactly what type of rotation this is, we can cleverly factor out a $\sqrt{5}$ to get $\left\{\begin{array}{l}x'=\sqrt{5}\left(\dfrac{1}{\sqrt{5}}x+\dfrac{2}{\sqrt{5}}y\right)\\ y'=\sqrt{5}\left(\dfrac{2}{\sqrt{5}}x-\dfrac{1}{\sqrt{5}}y \right)\end{array}\right.$

Now let $\theta$ be an angle satisfying $\sin\theta = \dfrac{2}{\sqrt{5}}$ and $\cos\theta = \dfrac{1}{\sqrt{5}}$ . Our transformation is now $\left\{\begin{array}{l}x'=\sqrt{5}\left(x\cos\theta+y\sin\theta\right)\\ y'=\sqrt{5}\left(x\sin\theta-y\cos\theta \right)\end{array}\right.$

Now it is clear what our transformation is: a rotation of angle $\theta$ and a scale up by a factor of $\sqrt{5}$ .

Once we are clear what we actually did to our graph, let's look at the new graph: $|x'|+|y'|=10$

This graph is simply a diamond, which has an area of $200$ .

Now applying our transformation in reverse, we have to scale each dimension down by $\sqrt{5}$ (rotations don't change area). This means the area is scaled down by $\sqrt{5}^2=5$ , so the area our original graph bounds is $200\div 5=\boxed{40}$

We can considerate all cases for each absolut value

Case 1: both positive

$x+2y+2x-y=10$

$\boxed{3x+y=10}$

Case 2: First positive and second negative

$x+2y-2x+y=10$

$\boxed{-x+3y=10}$

Case 3: First negative and second positive

$-x-2y+2x-y=10$

$\boxed{x-3y=10}$

Case 4: Both negative

$-x-2y+2x-y=10$

$\boxed{-3x-y=10}$

Solving every couple of ecuation, we get it that the graph is an square which vertex are ${(2,4),(4,-2),(-4,2), (-2,-4)}$ .

Then we find that a side of the square is $\sqrt{40} \therefore$ the area is $\boxed{40}$

By opening mod of the 2terms we get 4 linear equations in 2 variables.-: 3x+y=10. 3x+y=-10 x-3y=10.. x-3y=-10. So 1st and 2nd are parallel and perpendicular to 3rd and 4th So the figure is either square or rectangle Area=l*b where l&b are the distances between each pair of parallel lines. =20/root of 10 for both... Area=[20/root 10]^2=400