1. The problem states we have a line $l_1$ with equation $x + 2y = 4$. 2. We need to find line $l_2$ perpendicular to $l_1$ and passing through the origin. 3. First, rearrange $l_1$ into slope-intercept form $y = mx + b$: $$x + 2y = 4 \Rightarrow 2y = -x + 4 \Rightarrow y = -\frac{1}{2}x + 2$$ 4. The slope of $l_1$ is $m_1 = -\frac{1}{2}$. 5. The slope of a line perpendicular to $l_1$ is the negative reciprocal of $m_1$: $$m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2$$ 6. Since $l_2$ passes through the origin $(0,0)$, its equation is: $$y = m_2 x + 0 = 2x$$ 7. So, the equation of the line $l_2$ perpendicular to $l_1$ and passing through the origin is $y=2x$.