Question bank — Parallel lines — equal slopes

False — if they also share the same $y$-intercept they are the same (coincident) line, meeting at infinitely many points, not "never meeting."

True in a plane for non-vertical lines — never-meeting forces equal steepness; but note two vertical lines also never meet yet have undefined (not "equal number") slopes.

True — both are vertical, so they run side by side forever; their slopes are both undefined, which is the vertical version of "same direction."

True — both are horizontal with slope $0$; equal slopes ($0=0$), different intercepts, so they never meet.

Depends on convention — geometrically a line trivially has the same direction as itself, but "parallel" usually means distinct lines, so most textbooks say no (that case is "coincident").

False — all three ratios equal means the equations are scalar multiples, i.e. the same line. Parallel needs the $c$-ratio to differ.

In spirit yes: parallel is $m_1=m_2$, perpendicular is $m_1{m}_2=-1$; they can never hold at once for two real slopes since $m^2=-1$ has no real solution.

True — a line's direction is the same whether measured "forwards" or "backwards," and $\tan \theta$ repeats every $180^{\circ }$, so equal $\tan$ (equal slope) still holds.

False — $2ax+2by+2c=0$ has slope $-\frac{2a}{2b}=-\frac{a}{b}$, identical to the original; it's literally the same line, so slope is unchanged.

Error: $-6/4$ equals $-3/2$. Both slopes are $-3/2$, so the lines are parallel (constants $7$ vs $7$ scale differently, so distinct).

Backwards — you must ==keep $a,b$== (they fix the direction/slope) and change only the constant: $ax+by+k=0$.

Error: for $x=4$ the run (x-change) is $0$, so $m=\frac{\text{rise}}{0}$ is undefined, not $0$. Zero slope is the horizontal case $y=c$.

Error: $L_2$ has slope $-\frac{-2}{3}=\frac{2}{3}$, same as $L_1$'s $-\frac{2}{-3}=\frac{2}{3}$. Multiplying an equation by $-1$ never changes its slope, so they are parallel.

Over-complication — each line makes one fixed angle $\theta$ with the x-axis; comparing those two single angles (${\theta }_1$ vs ${\theta }_2$) is enough. No multiple checks needed.

Error: equal slopes keep the vertical gap constant, not zero. The perpendicular distance is $\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$ — see Distance Between Two Parallel Lines.

Same slope means the same rise for the same run, so the vertical gap between the lines stays constant everywhere — a constant nonzero gap can never shrink to zero, so they never touch.

Slope is rise/run = opposite/adjacent on the angle triangle, and that ratio is exactly $\tan \theta$; $\sin$ and $\cos$ involve the hypotenuse, which slope doesn't care about.

To go from ${\theta }_1={\theta }_2$ back to $m_1=m_2$ and the reverse, we need each angle to give a unique tan value on $(-90^{\circ },90^{\circ })$; one-to-one guarantees equal tangents force equal inclinations.

A vertical line has run $=0$, making $m$ undefined; "undefined $=$ undefined" isn't an equation of numbers, so vertical parallels must be spotted by inspection instead.

The slope $-a/b$ depends only on $a$ and $b$; fixing them fixes the direction, and changing the constant $c$ only slides the line without tilting it — see General Equation of a Line ax+by+c=0.

Coincident means the two equations describe one identical line (they agree at every point), whereas parallel means same direction but distinct — parallel is the weaker requirement, matching only $a,b$ ratios.

No — the x-axis is horizontal (slope $0$) and the y-axis is vertical (undefined slope); different directions, and they meet at the origin, so they're perpendicular, not parallel.

Never — one has slope $0$, the other undefined; different directions, and they always intersect at exactly one point.

Same line — equal slope and the shared point $(0,0)$ (same intercept) forces them to coincide, so they meet everywhere, not "never."

Every horizontal line $y=c$ for any constant $c$ (except $y=$ its own intercept, which would be identical); all share slope $0$.

No — equal slope and equal y-intercept means identical $y=mx+c$, so they're the same line; distinct parallels must differ in intercept.

$\tan \theta \to \infty$, so the slope blows up toward the vertical (undefined) case; the whole family of parallels becomes near-vertical lines, handled by inspection rather than the number rule.