Parallel lines — equal slopes: Worked examples

Before anything, one reminder in plain words. The slope of a line is the number $m = \frac{rise}{run} = tan θ,$ where "rise" is how far you go up, "run" is how far you go across, and $θ$ is the angle the line tilts up from the flat horizontal $x$ -axis (see Slope of a Line). Two lines are parallel — never meeting — exactly when they tilt by the same amount, i.e. $m_{1} = m_{2}$ .

The scenario matrix

Every parallel-lines question falls into one of these cells. The worked examples that follow are each tagged with the cell they cover.

Cell	Situation	What's tricky	Example
A	Both slopes positive	routine sanity check	Ex 1
B	Both slopes negative	sign of $- a / b$	Ex 2
C	Slope zero (horizontal)	is $0$ a real slope?	Ex 3
D	Slope undefined (vertical)	rule $m_{1} = m_{2}$ breaks	Ex 4
E	Same slope, same line	parallel vs coincident trap	Ex 5
F	Find an unknown $k$ for parallelism	solve an equation	Ex 6
G	Distance between two parallels	needs matching $a, b$	Ex 7
H	Word problem (real world)	translate words → slopes	Ex 8
I	Exam twist: parallel + a point, mixed forms	combine skills	Ex 9

Figures accompany the geometric cells (A, C, D, G).

Worked examples

Worked example Ex 1 — Cell A: both slopes positive

Are $L_{1} : 4 x - 2 y + 3 = 0$ and $L_{2} : 6 x - 3 y - 5 = 0$ parallel?

Forecast: both have the $x$ -term bigger than the $y$ -term with the same sign pattern — guess parallel or not before reading on.

Slope of $L_{1}$ . Use $m = - \frac{a}{b}$ (from General Equation of a Line ax+by+c=0): here $a = 4, b = - 2$ , so $m_{1} = - \frac{4}{- 2} = 2$ . Why this step? $- a / b$ is the fastest route from general form to slope — no rearranging needed.
Slope of $L_{2}$ . $a = 6, b = - 3$ , so $m_{2} = - \frac{6}{- 3} = 2$ . Why this step? Same tool, applied to the second line so we can compare.
Compare. $m_{1} = m_{2} = 2$ , both positive — the lines tilt up equally steeply. Constants differ ( $3$ vs $- 5$ ), so they are distinct parallel lines. ✅ Why this step? Equal slope ⇒ parallel; different constant ⇒ not the same line.

Verify: Rewrite each as $y = m x + c$ : $L_{1} : y = 2 x + \frac{3}{2}$ , $L_{2} : y = 2 x - \frac{5}{3}$ . Same $2$ , different intercept — confirms parallel. Look at the figure: two up-tilting lines that stay a fixed gap apart.

Worked example Ex 2 — Cell B: both slopes negative

Are $L_{1} : 3 x + 5 y - 8 = 0$ and $L_{2} : 9 x + 15 y + 2 = 0$ parallel?

Forecast: $L_{2}$ looks like $3 \times L_{1}$ except for the constant — guess.

Slope of $L_{1}$ : $a = 3, b = 5 \Rightarrow m_{1} = - \frac{3}{5}$ . Why this step? Both $a, b$ positive, so $- a / b$ is negative — the line tilts down. We must respect that sign.
Slope of $L_{2}$ : $a = 9, b = 15 \Rightarrow m_{2} = - \frac{9}{15} = - \frac{3}{5}$ . Why this step? Reduce the fraction; $9/15 = 3/5$ so the slopes match.
Compare: $m_{1} = m_{2} = - \frac{3}{5}$ . Both negative, equal ⇒ parallel. Check the constant ratio to be safe: $\frac{3}{9} = \frac{5}{15} = \frac{1}{3}$ but $\frac{- 8}{2} = - 4 \neq = \frac{1}{3}$ , so not the same line. Parallel ✅. Why this step? Matching $a, b$ ratios could hide a coincident line; the $c$ ratio being different rules that out.

Verify: $L_{1} : y = - \frac{3}{5} x + \frac{8}{5}$ , $L_{2} : y = - \frac{3}{5} x - \frac{2}{15}$ . Equal slope $- \frac{3}{5}$ , different intercept — parallel confirmed.

Worked example Ex 4 — Cell D: vertical lines (undefined slope)

Are $x = - 2$ and $3 x = 9$ parallel?

Forecast: vertical lines. The rule $m_{1} = m_{2}$ uses numbers — do vertical lines even have a slope number?

Try the slope formula. For $x = - 2$ , run is always $0$ : $m = \frac{rise}{0}$ = undefined (dividing by zero). Same for $3 x = 9 \Rightarrow x = 3$ . Why this step? To show why the numeric rule fails here — you cannot write " $m_{1} = m_{2}$ " when neither slope is a number.
Handle by inspection. Both lines are of the form $x = constant$ , so both point straight up. Same direction ⇒ parallel. ✅ Why this step? The equal-slope theorem was proved only for non-vertical lines; vertical lines need this direct geometric check.
Distinct? Constants $- 2 \neq = 3$ ⇒ two different verticals, genuinely parallel (not the same line). Why this step? Same "same-line vs parallel" caution applies here too.

Verify: Both make a $9 0^{\circ}$ angle with the $x$ -axis; $tan 9 0^{\circ}$ is undefined, matching "no slope number." Figure shows two vertical rails a fixed horizontal gap apart.

Worked example Ex 5 — Cell E: same slope BUT same line (the trap)

Are $L_{1} : x - 2 y + 3 = 0$ and $L_{2} : 3 x - 6 y + 9 = 0$ parallel?

Forecast: slopes will match — but be careful, "equal slope" is not the whole story.

Slopes. $m_{1} = - \frac{1}{- 2} = \frac{1}{2}$ ; $m_{2} = - \frac{3}{- 6} = \frac{1}{2}$ . Equal. Why this step? Equal slope is necessary for parallel — but we must check it isn't the same line.
Check all three ratios. $\frac{a _{1}}{a _{2}} = \frac{1}{3}$ , $\frac{b _{1}}{b _{2}} = \frac{- 2}{- 6} = \frac{1}{3}$ , $\frac{c _{1}}{c _{2}} = \frac{3}{9} = \frac{1}{3}$ . All three equal. Why this step? Parallel needs $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}$ ; if the $c$ ratio matches too, it's the same line.
Conclude. $L_{2} = 3 \times L_{1}$ exactly — they are the same (coincident) line, NOT parallel. ❌ parallel. Why this step? Coincident lines meet at every point; parallel means never meet — opposites in the "meeting" sense.

Verify: Both simplify to $y = \frac{1}{2} x + \frac{3}{2}$ — identical intercept too. One line drawn twice.

Worked example Ex 7 — Cell G: distance between two parallels

Find the perpendicular distance between $L_{1} : 3 x - 4 y + 6 = 0$ and $L_{2} : 6 x - 8 y - 5 = 0$ .

Forecast: distance formula needs matching $a, b$ . Are they matching yet? Guess what fix you'll do first.

Confirm parallel & match coefficients. $L_{2}$ has $a, b = 6, - 8 = 2 \times (3, - 4)$ . Divide $L_{2}$ by $2$ : $3 x - 4 y - \frac{5}{2} = 0$ . Now both share $a = 3, b = - 4$ . Why this step? The formula $\frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$ only works when the normal direction $(a, b)$ is identical in both — otherwise $c_{1} - c_{2}$ isn't measured along the same ruler (see Distance Between Two Parallel Lines).
Apply the formula. $c_{1} = 6, c_{2} = - \frac{5}{2}$ : $d = \frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}} = \frac{6 - ( - \frac{5}{2} )}{3 ^{2} + ( - 4 ) ^{2}} = \frac{\frac{17}{2}}{25} = \frac{17}{10} = 1.7.$ Why this step? Numerator = gap in constants along the shared normal; denominator = length of that normal vector, which rescales the gap to true distance.

Verify: Pick a point on $L_{1}$ , say where $y = 0$ : $3 x + 6 = 0 \Rightarrow x = - 2$ , point $(- 2, 0)$ . Distance from $(- 2, 0)$ to $L_{2}$ (in matched form $3 x - 4 y - \frac{5}{2} = 0$ ): $\frac{∣3 ( - 2 ) - 4 ( 0 ) - \frac{5}{2} ∣}{5} = \frac{∣ - 6 - \frac{5}{2} ∣}{5} = \frac{17/2}{5} = 1.7$ . ✅ Same. Figure shows the perpendicular gap.

Worked example Ex 8 — Cell H: word problem (real world)

Two straight roads are drawn on a map. Road $P$ climbs from town $(0, 1)$ to $(4, 4)$ . Road $Q$ passes through $(1, 0)$ and must never cross Road $P$ . Road $Q$ also passes through $(5, y_{0})$ . Find $y_{0}$ .

Forecast: "never cross" is a hidden word for parallel. Once you spot that, the rest is slope-matching.

Slope of Road $P$ . $m_{P} = \frac{4 - 1}{4 - 0} = \frac{3}{4}$ . Why this step? Rise over run from its two given points — the direction the road heads.
Translate "never cross" → parallel → equal slope. So $m_{Q} = \frac{3}{4}$ too. Why this step? On flat ground, two roads never meet ⇔ same tilt (they share direction forever).
Use $Q$ 's known point $(1, 0)$ with slope $\frac{3}{4}$ (Point-Slope Form of a Line): $y - 0 = \frac{3}{4} (x - 1)$ . Why this step? Point + slope pins down the line uniquely.
Plug in $x = 5$ : $y_{0} = \frac{3}{4} (5 - 1) = \frac{3}{4} \cdot 4 = 3$ . Why this step? We want $Q$ 's height at $x = 5$ ; substitute.

Verify: Slope of $Q$ from $(1, 0)$ to $(5, 3)$ : $\frac{3 - 0}{5 - 1} = \frac{3}{4} = m_{P}$ . ✅ Roads stay parallel, never cross.

Worked example Ex 9 — Cell I: exam twist (mixed forms + a point)

A line is parallel to $2 x + 3 y - 6 = 0$ and passes through the intersection of $x - y = 1$ and $x + y = 5$ . Find its equation.

Forecast: two mini-problems glued together — first find a point, then build a parallel line through it.

Find the intersection point. Solve $x - y = 1$ and $x + y = 5$ : adding gives $2 x = 6 \Rightarrow x = 3$ ; then $y = 5 - 3 = 2$ . Point $(3, 2)$ . Why this step? "Passes through the intersection" means we need that meeting point's coordinates first.
Target slope. $2 x + 3 y - 6 = 0$ has $m = - \frac{2}{3}$ . Why this step? Parallel ⇒ our line copies this slope.
Shortcut form (General Equation of a Line ax+by+c=0): keep $a, b$ , change constant → $2 x + 3 y = k$ . Plug $(3, 2)$ : $2 (3) + 3 (2) = 6 + 6 = 12$ . So $k = 12$ . Why this step? "Keep $a, b$ , change $c$ " instantly makes a parallel line; the point fixes the constant.
Answer: $2 x + 3 y = 12$ .

Verify: Slope of $2 x + 3 y = 12$ is $- \frac{2}{3}$ (matches target) and $2 (3) + 3 (2) = 12$ ✅ (passes through $(3, 2)$ ).

Common traps this matrix exposes

Active Recall

Recall Which cell is each? (forecast the tag first)

$y = 4$ and $y = 7$ → ::: Cell C (zero slope, parallel)
$x = - 2$ and $x = 3$ → ::: Cell D (undefined slope, parallel by inspection)
$x - 2 y + 3 = 0$ and $3 x - 6 y + 9 = 0$ → ::: Cell E (same line, NOT parallel)
"roads that never cross" → ::: Cell H (word problem = parallel)

Recall Numeric answers to re-derive

Distance in Ex 7? ::: $1.7$
$k$ in Ex 6? ::: $- \frac{8}{3}$
$y_{0}$ in Ex 8? ::: $3$
Ex 9 final line? ::: $2 x + 3 y = 12$

Connections

Parallel Lines — Equal Slopes (parent)
Slope of a Line
Point-Slope Form of a Line
General Equation of a Line ax+by+c=0
Distance Between Two Parallel Lines
Angle Between Two Lines
Perpendicular Lines — Product of Slopes = −1

Worked examples — Parallel lines — equal slopes