Straight lines: slope from general form, and signs from a sketch

6 min read·Updated 5 September 2026

A straight line arrives in Paper 2 in one of two disguises. Either you are given its equation and asked for a geometric fact — slope, intercept, a perpendicular — or you are shown a sketch and asked what its equation's coefficients must satisfy. The first is mechanical once you have the right formula. The second is where the marks go.

Misconception 1: slope from the general form

Written as $ax + by + c = 0$, the line's slope is

not $a/b$ and not $-b/a$. The minus sign is not a convention to memorise; it falls straight out of rearranging into $y = mx + d$:

The same rearrangement hands you the $y$-intercept, $-c/b$, which is the other value people quote with the wrong sign. If you ever doubt either, spend the four seconds rearranging rather than recalling.

Misconception 2: half of "negative reciprocal"

Two perpendicular lines have slopes whose product is $-1$, so the perpendicular slope is the negative reciprocal:

That is two operations — flip, then negate. Applying one of them is the single commonest error in the topic, and it is invisible in the working because each half is a legitimate-looking manipulation. A slope of $\tfrac{2}{3}$ has perpendicular slope $-\tfrac{3}{2}$; the two near misses, $\tfrac{3}{2}$ and $-\tfrac{2}{3}$, are exactly the numbers a well-built option list will contain.

The check costs nothing: multiply your two slopes. If the product is not $-1$, you did one operation.

Misconception 3: assuming the coefficients are positive

Given a sketch and the equation $ax + by + c = 0$, the question is usually which of $a$, $b$, $c$ are positive. The error is to reason as though $b \gt 0$ and read everything else off that assumption — which is right exactly half the time and gives no signal when it is not.

Reason from the two things a sketch really tells you, the sign of the slope and the sign of the $y$-intercept, and translate each into a statement about a ratio:

• slope $-a/b$ negative $\Rightarrow$ $a$ and $b$ have the same sign;
• intercept $-c/b$ positive $\Rightarrow$ $c$ and $b$ have opposite signs.

Ratios are all a sketch can determine. Any question that has a unique answer must have pinned one coefficient's sign some other way — find where it did that before choosing.

A worked example

Consider the line $2x - 3y + 6 = 0$.

Rearranging: $3y = 2x + 6$, so $y = \tfrac{2}{3}x + 2$. The slope is $\tfrac{2}{3}$ and the $y$-intercept is $2$.

Reading the slope straight off the coefficients as $\tfrac{2}{-3}$ gives $-\tfrac{2}{3}$ — the right magnitude, the wrong sign, and a line sloping the wrong way. Reading it as $-\tfrac{3}{2}$ inverts it as well.

Now the perpendicular through the same intercept:

Check: $\tfrac{2}{3} \times \left(-\tfrac{3}{2}\right) = -1$. Had we written $\tfrac{3}{2}$, the product would have been $1$ and the error would have announced itself in one multiplication.

Check yourself

Rearrange, don't recall — four seconds of algebra beats a half-remembered sign. And after every perpendicular slope, multiply the two slopes and confirm you get $-1$.

The drill below generates new coefficients each time, including negative ones, so the "assume positive" shortcut cannot survive.

Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.