Slicing a Square

The area of the square is $1$ . Each of the colored regions, then, must have an area of $\frac{1}{3}$ . The area of each triangle is given by,

$A = \frac{base \cdot height}{2} = \frac{1}{3}$

So, the height of the bottom one and the width of the top one must be $\frac{2}{3}$ .

So, the slope of the lower line, $m1$ , is $\frac{2}{3}$ and the slope of the upper one, $m2$ , is $\frac{3}{2}$ .

So, $m_1 + m_2 = \frac{2}{3} + \frac{3}{2} = \frac{13}{6}$

$13+6 = \boxed{19}$