Parallel and perpendicular lines

Comparing the slopes of two lines tells you whether they are parallel or perpendicular.

Two lines with equal slopes are parallel and never meet. For example $y = 2x - 1$ and $y = 2x + 3$ both have slope $2$: they rise at the same rate as $x$ increases, so they stay a fixed distance apart and never cross. Indeed, setting $2x - 1 = 2x + 3$ gives $-1 = 3$, which no $x$ satisfies, so there is no solution.

Two lines whose slopes multiply to $-1$ meet at a right angle. If $m$ is the slope perpendicular to a line of slope $2$, then $2m = -1$, so $m = -\dfrac{1}{2}$. The line with slope $-\dfrac{1}{2}$ through $(1, 1)$ is $y = \dfrac{3 - x}{2}$, and it crosses $y = 2x - 1$ at $(1, 1)$ at a right angle.

In summary, for two lines with slopes $m_1, m_2$: they are parallel if $m_1 = m_2$, and perpendicular if $m_1 m_2 = -1$. The graph shows two parallel lines of slope $2$ and one line perpendicular to them, with the large dot at their right-angle crossing $(1, 1)$.