Parallel lines share a slope. Perpendicular slopes multiply to −1. The SAT's whole job is to hide the slope somewhere you have to dig it out of.
What is the value of the slope?
In the xy-plane, line k is perpendicular to the line given by 4x + 5y = 20. What is the slope of line k?
Solve for y first: 5y = −4x + 20, so y = −(45)x + 4 and the given line has slope −45. Perpendicular means flip the fraction and switch the sign, giving 54, and (−45)(54) = −1 confirms it. B, −45, is the given line's own slope — the answer to “parallel”, not “perpendicular”. A flips without switching the sign, and C switches the sign without flipping. Each of those does half the job, and each has its own choice sitting there waiting.
1
Parallel lines have equal slopes — that is the entire rule.
In the xy-plane two distinct lines are parallel exactly when m₁ = m₂. The intercepts are free to be anything, as long as they are not also equal.
y = 3x − 7 ∥ y = 3x + 2
same m = 3, so they never meet
y = 3x − 7 and y = 3x − 7
same slope AND same intercept — one line, not two
y = 3x − 7 and y = 3.5x − 7
not parallel; they cross at (0, −7)
parallel → m₁ = m₂b₁ ≠ b₂no solution as a system
That last chip is worth holding on to. A system of two linear equations has no solution precisely when the lines are parallel, which is the same fact wearing a different hat.
2
Perpendicular lines: flip the fraction and switch the sign.
m₁ · m₂ = −1. The second slope is the negative reciprocal of the first — both moves, every time.
2 → −12
a whole number is a fraction over 1
−45 → 54
flip to 54, and the minus becomes a plus
−1 → 1
flipping 11 changes nothing; only the sign moves
(−45)(54) = −2020 = −1
the test, done in one line
Half the work is a wrong answer with its own choice
In the opening problem the given slope was −45. Flip only and you get −54; switch the sign only and you get 45; do neither and you get −45 back, which is the parallel slope. All three were printed as choices A, C and B. Say the two moves out loud — flip, then switch — and check the product is −1 before you commit.
3
Standard form hides the slope. Solve for y before you compare anything.
The SAT loves to write one line as y = mx + b and the other as Ax + By = C, so the two are never in the same shape.
4x + 5y = 20
the number in front of x is 4, but the slope is not 4
5y = −4x + 20
move the x-term across
y = −(45)x + 4
divide every term by 5 — now m is visible
2x − 3y = 12 → y = (23)x − 4
dividing by a negative flips both signs
Shortcut worth memorising: for Ax + By = C the slope is −AB. Check it — 4x + 5y = 20 gives −45, and 2x − 3y = 12 gives −2−3 = 23. Both match the long way.
6x + 9y = 18 → m = −69 = −23
reduce the fraction before comparing
2x + 3y = 30 → m = −23
same slope, so these two are parallel
Two standard-form lines are parallel when A₁A₂ = B₁B₂ but the constants do not follow the same ratio: 62 = 93 = 3, while the constants give 1830 = 35, not 3. If the constants had matched too, the two equations would be the same line written twice.
4
What a perpendicular pair actually looks like.
Here are y = 2x − 1 and y = −x2 + 4 on one grid. The axes are drawn at the same scale — one unit across is exactly one unit up — which is the only way a right angle looks like a right angle.
blue: from (1, 1) to (2, 3) — up 2, across 1
m = 2
gold: from (0, 4) to (2, 3) — down 1, across 2
m = −12
2 × (−12) = −1
perpendicular, as the corner square says
2x − 1 = −x2 + 4 → 2.5x = 5 → x = 2, y = 3
and that is the marked crossing point
A drawing can lie about a right angle
Stretch the horizontal axis and the same two lines will look like they meet at 60°. That is why the SAT asks you to read slopes off labelled points rather than to judge the corner by eye — and why, on a graph with unequal axis scales, “looks perpendicular” proves nothing.
5
Through a given point: get the slope, then let the point pin down b.
Two questions, one procedure. Both use 4x + 5y = 20, whose slope we already found to be −45.
parallel, through (5, 2): m = −45
parallel copies the slope unchanged
2 = −(45)(5) + b = −4 + b
put the point into y = mx + b
b = 6, so y = −(45)x + 6
check: −(45)(5) + 6 = 2 ✓
perpendicular, through (4, −1): m = 54
flip and switch −45
−1 = (54)(4) + b = 5 + b
the 4 and the denominator cancel — points are chosen kindly
b = −6, so y = (54)x − 6
check: (54)(4) − 6 = −1 ✓
1. rearrange to find m2. keep it or flip-and-switch it3. substitute the point for b
Where the calculator earns its keep
Graph the original line and your answer in Desmos, then type the point as (4, −1) and check the dot sits on your line. Use Zoom Square first, so the axes share a scale and a right angle appears as one. If a question offers four candidate equations, graphing all four and looking for the one that passes through the given point is often faster than algebra.
6
The parameter version: the slope is a letter and the relationship is the clue.
“Line t is defined by y = ax + 3, where a is a constant. Line t is parallel to the line given by 2x − 8y = 5. What is the value of a?”
2x − 8y = 5 → −8y = −2x + 5
isolate the y-term
y = (14)x − 58
divide by −8; −2/−8 reduces to 14
parallel → a = 14
the intercept 3 never enters the calculation
perpendicular instead → a = −4
flip 14 to 4, switch the sign
One word in the stem — parallel or perpendicular — is the whole difference between 14 and −4, and both numbers will be among the choices.
Pair
Slopes
Verdict
y = 2x − 3 and y = 2x + 5
2 and 2
parallel
y = 2x − 3 and y = −x2 + 4
2 and −12
perpendicular
y = 4 and x = 2
0 and undefined
perpendicular — but no product exists
Horizontal and vertical break the formula
A horizontal line has slope 0 and a vertical line has no slope at all, so 0 × (undefined) is not −1 — it is not anything. The pair is still perpendicular; recognise it by sight. Likewise, “parallel to the y-axis” means a line of the form x = c, which can never be written as y = mx + b.
✓
Check: multiply the slopes, then test the point.
the opener(−45)(54) = −1perpendicular confirmed ✓
the parallel line−(45)(5) + 6 = 2it really passes through (5, 2) ✓
the perpendicular line(54)(4) − 6 = −1it really passes through (4, −1) ✓
Remember the flow
Solve for y · Parallel keeps m · Perpendicular flips AND switches · Point fixes b
If the product of the slopes is not −1, it is not a perpendicular pair.
Now you try
Practice problem
Which equation defines line m?
Line ℓ is defined by 2x − 7y = 14. In the xy-plane, line m is perpendicular to line ℓ and passes through the point (4, 3). Which equation defines line m?
Rearrange first: −7y = −2x + 14, so y = (27)x − 2 and line ℓ has slope 27. Flip and switch for the perpendicular slope, −72, then use the point: 3 = −(72)(4) + b = −14 + b, so b = 17 and y = −(72)x + 17. A keeps the slope 27, which builds the parallel line through (4, 3) instead. C flips but forgets the sign, giving 3 = 14 + b and b = −11. D switches the sign without flipping, giving 3 = −87 + b and b = 297. Only one of the four passes through (4, 3) with a slope whose product with 27 is −1.
Hint: do not touch the point until the slope is settled. Get line ℓ into y = mx + b, decide what perpendicular does to that slope, and only then substitute (4, 3) to find b.
