Coordinates, Straight Lines and Gradient — IGCSE Mathematics 0580

1 · Coordinates and midpoints 2 · What gradient means 3 · Gradient between two points 4 · y = mx + c 5 · Parallel and perpendicular 6 · Length and drawing 7 · Mixed set

What the check actually found

In this strand — coordinates, straight lines and gradient — you were secure all the way to step 4 and the first gap opened at step 5. Plotting points, reading a grid, negative coordinates: all fine. What broke is the point where the grid stops being a picture and starts being a calculation — measuring steepness, and turning a line into an equation.

What it maps onto in the syllabus

Code	Topic	Risk
E3.1	Coordinates in two dimensions	Secure — revised here briefly
E3.2	Drawing linear graphs	Section 6
E3.3	Gradient of a straight line	High non-calculator risk
E3.5	Equations of linear graphs	High non-calculator risk
E3.6	Parallel lines	High non-calculator risk
E3.4	Length of a line segment	Section 6

Three of the six are flagged high non-calculator risk, which matters because Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. Everything here is by hand, every line shown, and nowhere are you sent to a machine.

The finding that shapes this whole guide

The check picked up one error that recurred right across different strands: sign errors. This is the worst possible topic to carry that habit into. Gradient is a subtraction of two numbers that are routinely negative, divided by another subtraction of two numbers that are routinely negative. A dropped minus sign does not make the answer slightly wrong — it turns an uphill line into a downhill one. Sections 2, 3 and 5 attack it directly.

The part of the check that went right

Number sense and the four operations: solid on all seven rungs. No gap anywhere. Arithmetic is not your problem, and this is a strand where that counts for a lot. The numbers in gradient work are small — you will divide 9 by 3, or 10 by 5, and almost never anything worse. There is no hidden arithmetic tax waiting for you here.

Being secure to step 4 also means the picture is already in place: you can read a coordinate grid, plot a point and handle negative coordinates. The repair is one storey up, and it is a short one.

How this guide works

Every method appears four times and hands you less each time.

Worked in full — every line, each with the reason it was done.
Last step is yours — finish the final line before pressing the button.
Last two are yours — the same, with less scaffolding.
All yours — type an answer and have it marked.

Section 1 sits below your break and is deliberately short. Sections 2 and 3 are the long ones, because that is where step 5 actually failed and where the sign errors live.

step 3–41 · Coordinates and midpoints▼

▶ Watch: E3.1 Coordinates · E3.4 Length and midpoint

Short hand-worked explanations for exactly this sub-topic. Opens on YouTube in a new tab. Watch one, then come straight back and try the questions below — watching without testing yourself feels like learning but is not.

A coordinate is always written (across, up) — x first, then y. That order never changes, and it is the one thing in this section worth saying out loud every time you write a bracket. Along the corridor, then up the stairs.

Four quadrants, a scale, and a missing corner

The axes cut the page into four quadrants, numbered anticlockwise from the top right. A negative x means left of the y-axis; a negative y means below the x-axis.

quadrant	x	y	example
first (top right)	+	+	(3, 2)
second (top left)	−	+	(−3, 2)
third (bottom left)	−	−	(−3, −2)
fourth (bottom right)	+	−	(3, −2)

Read the scale before you count squares. On an exam grid one square can be 2 units, 5 units or 0.5 of a unit, and the two axes can have different scales. Read the numbers printed on each axis first.

A missing corner of a parallelogram. Opposite sides of a parallelogram are equal and parallel, so the move from B to C is exactly the same as the move from A to D. Find the move, then repeat it.

Worked in full

ABCD is a parallelogram. A(−2, −1), B(3, 1) and C(5, 6). Find the coordinates of D.

B → C: across 5 − 3 = 2, up 6 − 1 = 5

The move along one side. Subtract the start from the end, x and y separately.

A → D is the same move: (+2, +5)

BC and AD are opposite sides, so they are equal and parallel.

D = (−2 + 2, −1 + 5) = (0, 4)

Add the move to A.

check: A → B is (5, 2) and D → C is (5 − 0, 6 − 4) = (5, 2) ✓

The other pair of opposite sides must match too.

All yours

On a grid, one square across is 0.5 units and one square up is 10 units. P is 3 squares right and 2 squares up from the origin. Write down P, like (3,4).

Q is 3 units to the left of (1, 2) and 4 units below it. Write down Q, like (3,4).

ABCD is a parallelogram. A(1, 1), B(6, 2) and C(9, 6). Find D, like (3,4).

The midpoint of two points is the point exactly halfway between them. Halfway across, and halfway up. Halfway between two numbers is their average, so:

midpoint = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )

The mistake: two of them, and both are sign-related.
• Adding the coordinates and forgetting to halve. (−4 + 6) is 2, and the midpoint x is 1.
• Subtracting instead of adding, because gradient work uses subtraction and the two get mixed up. Midpoint adds. Gradient subtracts. Keep them apart: a midpoint has to lie between the two points, so if your answer is outside them, you subtracted.

Worked in full

Find the midpoint of A(2, 3) and B(8, 11)

x: (2 + 8) ÷ 2 = 10 ÷ 2 = 5

Average the two x values. 5 lies between 2 and 8, as it must.

y: (3 + 11) ÷ 2 = 14 ÷ 2 = 7

Average the two y values in exactly the same way.

midpoint = (5, 7)

Write it as a coordinate pair, x first.

check: 5 is between 2 and 8, 7 is between 3 and 11 ✓

A midpoint that is not between the two points is not a midpoint. This check is free.

Last step is yours

Find the midpoint of (1, 4) and (7, 10)

x: (1 + 7) ÷ 2 = 4

Average the x values.

y: (4 + 10) ÷ 2 = 7

Average the y values.

midpoint = (4, 7)

Both values lie between the originals, so it is sound.

Last two are yours

Find the midpoint of (−3, 5) and (7, −1)

x: (−3 + 7) ÷ 2

Adding a negative to a positive: start at −3 and count up 7. Do not let the minus turn this into a subtraction of 10.

= 4 ÷ 2 = 2

−3 + 7 = 4, and half of 4 is 2. And 2 does sit between −3 and 7.

y: (5 + −1) ÷ 2 = 4 ÷ 2 = 2, so the midpoint is (2, 2)

5 + (−1) = 4. Adding a negative is subtracting.

All yours

Two points: (−6, −2) and (4, 8).

Find the midpoint. Write it like (3,4).

Working backwards: if M is the midpoint of A and B, and you know A and M, then M is as far from B as it is from A. Count the step from A to M, then repeat it. From A(1, 2) to M(4, 6) is 3 across and 4 up, so B is another 3 across and 4 up: B(7, 10).

Find the midpoint of (2, 7) and (10, 1). Write it like (3,4).

Find the midpoint of (−5, 3) and (1, −9). Write it like (3,4).

M(4, 6) is the midpoint of A(1, 2) and B. Find B. Write it like (3,4).

Which quadrant contains the point (−3, −7)? Answer with a number 1, 2, 3 or 4, counting anticlockwise from the top right.

step 4–52 · What gradient actually means▼

▶ Watch: E3.2 Drawing linear graphs

Do not start from a formula. Start from a hill. Gradient is a number that answers one question: for every one step you take across, how far do you go up? A gradient of 3 means three up for one across — steep. A gradient of ¹⁄₂ means half a step up for one across — gentle. A gradient of 0 means flat.

You measure it by drawing a right-angled triangle underneath the line, with one side horizontal and one vertical. The horizontal side is the run. The vertical side is the rise. Then:

gradient = rise ÷ run

The mistake, part one: computing run over rise. This gets the answer upside down, and the giveaway is that steep lines come out with small gradients. Anchor it with the words: gradient answers “how much UP”, so up is on top. Rise is on the rise.

The mistake, part two — and this is your one: losing the sign. A line that falls as you read left to right has a negative gradient, and a great many marks are lost by writing 2 where the answer was −2. The picture is worth more than any rule here: point at the line with your finger, start on the left, and move right. Going uphill is positive. Going downhill is negative. Do that check before you calculate, then check the sign of your answer against it afterwards.

Two special cases you must simply know. A horizontal line has zero rise, so its gradient is 0 ÷ run = 0. A vertical line has zero run, and dividing by zero is not allowed, so its gradient is undefined — the correct answer is to say so, not to write 0.

Worked in full

A line passes through (−2, −2) and (2, 4). Find its gradient from the triangle.

reading left to right, the line rises

Direction first. The answer must come out positive, so a negative answer means an error.

run: from x = −2 to x = 2 is 4 across

Count the squares along the bottom of the triangle.

rise: from y = −2 to y = 4 is 6 up

Count the squares up the side. From −2 to 0 is 2, and 0 to 4 is 4, so 6 altogether.

gradient = rise ÷ run = 6 ÷ 4 = 1.5

Up on top. Leave it as 1.5 or as 3/2 — both are acceptable.

positive, as predicted ✓

The sign matches the picture, so the answer is safe.

Last step is yours

A line goes 3 across and 12 up. Find its gradient.

run = 3, rise = 12, and the line rises

Up means positive.

gradient = 12 ÷ 3 = 4

Rise on top. Not 3 ÷ 12 = 0.25 — that would say the line was gentle, and 12 up for 3 across is steep.

Last two are yours

A line falls 8 down over 2 across. Find its gradient.

reading left to right the line falls, so the answer is negative

Decide the sign before touching the numbers. This is the habit that fixes sign errors.

rise = −8 (falling counts as a negative rise)

Down 8 is the same as up −8. The minus belongs on the rise, not on the run.

gradient = −8 ÷ 2 = −4

Negative, as the picture demanded. Writing 4 here would be the exact error the check flagged.

All yours

A line goes 6 across and falls 3 down.

What is its gradient? Write it like -1/2.

A line goes 5 across and 15 up. What is its gradient?

A line goes 4 across and falls 10 down. What is its gradient? Write it like -2.5.

What is the gradient of the line y = 4? Give a number.

A line goes 8 across and 2 up. What is its gradient? Write it like 1/4.

step 5 — the break3 · Gradient between two points▼

▶ Watch: E3.3 Gradient of linear graphs

Most exam questions do not give you a drawing. They give you two points and expect a number. The triangle from section 2 is still there — you just work out its sides by subtracting instead of counting squares.

gradient = (y₂ − y₁) ÷ (x₂ − x₁)

The rise is the difference in the y values. The run is the difference in the x values. That is the whole formula, and it is nothing more than the triangle written down.

The mistake — and this is the single biggest sign trap in the strand: mixing up which point is first on the top and which is first on the bottom. If you start with point B on the top, you must start with point B on the bottom too. Take A(1, 2) and B(4, 11):
• (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3 — B first on both. Correct.
• (2 − 11) ÷ (1 − 4) = −9 ÷ −3 = 3 — A first on both. Also correct.
• (11 − 2) ÷ (1 − 4) = 9 ÷ −3 = −3 — mixed. Wrong sign.
The order does not matter. Consistency does. Label your points, and write both subtractions in the same order without exception.

The habit that fixes it: write the two points one above the other before you start, top point first in both subtractions. Physically writing (y₂ − y₁) above (x₂ − x₁) with the same point subscripted the same way removes the choice, and removing the choice removes the error.

Worked in full

Find the gradient of the line through A(1, 2) and B(4, 11)

label: (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 11)

Do this in writing. It is the step that makes the mixed-order error impossible.

x goes from 1 up to 4, y goes from 2 up to 11 — the line rises

Predict the sign before calculating. Positive expected.

rise = y₂ − y₁ = 11 − 2 = 9

Point 2 first, as labelled.

run = x₂ − x₁ = 4 − 1 = 3

Point 2 first again. Same order, no exceptions.

gradient = 9 ÷ 3 = 3

Positive, as predicted in step 2 ✓

Now the version that costs marks: points with negative coordinates. The formula does not change at all. The only new thing is subtracting a negative, and subtracting a negative adds.

Worked in full

Find the gradient of the line through P(−2, 5) and Q(3, −5)

(x₁, y₁) = (−2, 5), (x₂, y₂) = (3, −5)

Label first, always.

y drops from 5 to −5 as x increases — the line falls

So the answer must be negative. Predicted before any arithmetic.

rise = −5 − 5 = −10

Not −5 − (−5). Read carefully: y₂ is −5 and y₁ is 5, so it is −5 take away 5.

run = 3 − (−2) = 3 + 2 = 5

Subtracting a negative adds. This is where the run most often comes out as 1 instead of 5.

gradient = −10 ÷ 5 = −2

Negative, matching the prediction. A negative divided by a positive is negative.

sanity: from P, going 1 right drops you 2 — and 5 rights drop you 10, landing on Q ✓

Walking the gradient back to the other point is the strongest check there is.

Last step is yours

Find the gradient of the line through (−4, −3) and (2, 6)

(x₁, y₁) = (−4, −3), (x₂, y₂) = (2, 6). y rises as x rises, so expect a positive answer.

Label, then predict.

rise = 6 − (−3) = 6 + 3 = 9

Subtracting a negative adds. Getting 3 here is the classic slip.

run = 2 − (−4) = 6, so gradient = 9 ÷ 6 = 3/2

Leave it as 3/2 or 1.5. Positive, as predicted.

Last two are yours

Find the gradient of the line through (5, 1) and (−1, 13)

(x₁, y₁) = (5, 1), (x₂, y₂) = (−1, 13)

Here point 2 is to the left of point 1. That is allowed — the formula does not care, as long as the order is consistent.

rise = 13 − 1 = 12

Point 2 on top, point 1 underneath it.

run = −1 − 5 = −6, so gradient = 12 ÷ −6 = −2

A positive divided by a negative is negative. And it should be: the point on the left is higher, so reading left to right the line falls.

All yours

A line passes through (−3, 7) and (1, −5).

Find its gradient.

Find the gradient of the line through (2, 3) and (6, 15).

Find the gradient of the line through (−1, 4) and (3, −4).

Find the gradient of the line through (−5, −2) and (−1, −10).

Find the gradient of the line through (4, −1) and (−2, 8). Write it like -3/2.

A line through (2, 5) and (2, 11). What is its gradient? Type undefined if it has none.

Find the gradient of the line through (−6, 3) and (2, 3). Give a number.

step 5–64 · y = mx + c▼

▶ Watch: E3.5 Equations of linear graphs

Every straight line that is not vertical can be written y = mx + c, where m is the gradient and c is the y-intercept — the y value where the line crosses the y-axis. Two numbers, and they describe the line completely: c says where it starts, m says how it tilts.

Reading a line off a graph: find c by looking at where the line cuts the y-axis — you read it, you do not calculate it. Then pick any two points on the line where it passes cleanly through grid corners, draw the triangle, and get m. Choose points far apart; a small triangle magnifies any misreading.

Worked in full

A line crosses the y-axis at 1 and passes through (2, 5). Find its equation.

c = 1, because that is the y value where it crosses the y-axis

The y-intercept is read straight off. No working needed.

the crossing point is (0, 1), so use (0, 1) and (2, 5) for the gradient

A y-intercept of 1 is a point on the line, and it is the most useful one because its x is zero.

m = (5 − 1) ÷ (2 − 0) = 4 ÷ 2 = 2

Same formula as section 3, same consistent order.

y = 2x + 1

Put m and c into the shape. Check by substituting x = 2: y = 4 + 1 = 5 ✓

Building the equation from a gradient and a point

More often you are handed m and one point, and c is missing. Substitute the point into y = mx + c and solve the one-step equation that results. This is where sign errors show up again, because c is very often negative.

Worked in full

A line has gradient 3 and passes through (2, 5). Find its equation.

y = 3x + c

Put in what you know. Only c is missing.

substitute x = 2, y = 5: 5 = 3 × 2 + c

The point lies on the line, so its coordinates must satisfy the equation.

5 = 6 + c

Work out the 3 × 2 before doing anything else.

c = 5 − 6 = −1

Subtract 6 from both sides. The answer is negative, and it stays negative — do not quietly write 1.

y = 3x − 1

Check: x = 2 gives 6 − 1 = 5 ✓. Always substitute the point back in.

Last step is yours

A line has gradient −2 and passes through (3, 1). Find its equation.

y = −2x + c

The gradient is negative, and that minus belongs in the equation from the start.

1 = −2 × 3 + c = −6 + c

−2 × 3 = −6. Negative times positive is negative.

c = 1 + 6 = 7, so the equation is y = −2x + 7

Adding 6 to both sides. Check: x = 3 gives −6 + 7 = 1 ✓

Last two are yours

Find the equation of the line through (1, −2) and (4, 7).

m = (7 − (−2)) ÷ (4 − 1) = 9 ÷ 3 = 3

Gradient first — you cannot find c without it. Subtracting the negative gave 9, not 5.

using (4, 7): 7 = 3 × 4 + c, so c = 7 − 12 = −5

Either point works. Use whichever has the friendlier numbers.

equation: y = 3x − 5

Check with the other point: x = 1 gives 3 − 5 = −2 ✓. Checking with the point you did not use is the strong check.

All yours

A line has gradient 4 and passes through (2, 3).

Find its equation. Write it like y=4x-5.

When the equation is not already in y = form

Examiners rarely hand you a tidy equation. They give you 2y = x + 6 or 3x + 2y = 12 and expect you to rearrange before reading anything off it.

The mistake: looking at 2y = x + 6 and reading off gradient 1 and intercept 6. Neither is right. The rule “m is the number in front of x” only holds when there is a single, bare y on the left. Until then, the numbers mean nothing. Rearrange first, read second.

Worked in full

Find the gradient and y-intercept of 2y = x + 6

the left side is 2y, not y — so nothing can be read off yet

Say this to yourself before looking at any of the numbers.

divide every term by 2: y = ^x⁄₂ + 3

Every term, including the 6. Dividing only the x term is the usual slip.

gradient m = ¹⁄₂, intercept c = 3

x/2 is the same as 0.5x, so the number in front of x is a half.

check with a point: x = 2 gives y = 4 in both forms

2y = 2 + 6 = 8 so y = 4; and 2/2 + 3 = 4. The rearrangement is sound.

Last step is yours

Find the gradient and y-intercept of 3x + 2y = 12

get the y term alone: 2y = −3x + 12

Subtract 3x from both sides. Moving it across the equals sign flips its sign to minus.

divide every term by 2: y = −³⁄₂x + 6, so m = −1.5 and c = 6

The gradient is negative. That minus came from moving 3x across, and dropping it is the sign error to watch for.

Last two are yours

Find the gradient of 4y − 8x = 20

add 8x to both sides: 4y = 8x + 20

Moving −8x across makes it +8x.

divide every term by 4: y = 2x + 5

8 ÷ 4 = 2 and 20 ÷ 4 = 5. All three terms, not just one.

gradient = 2

And the y-intercept is 5, if it were asked for.

All yours

The line 5y + 10x = 15.

What is its gradient?

What is the y-intercept of y = 7x − 4? Give a number.

A line has gradient −3 and passes through (1, 2). Find its equation. Write it like y=-3x+5.

Find the equation of the line through (0, −3) and (4, 5). Write it like y=2x-3.

What is the gradient of 6y = 3x − 12? Write it like 1/2.

Does the point (3, 8) lie on the line y = 2x + 2? Answer yes or no.

Other ways to write a line: ax + by = c, x = k and y = k

The same line can be written y = mx + c or ax + by = c. Exam questions often ask for the second form, with whole numbers. Four moves turn one into the other:

move	why
multiply every term to clear the fractions	the form wants whole numbers
bring the x and y terms to the left, the number to the right	that is the shape ax + by = c
make the x term positive (multiply every term by −1 if needed)	the usual way to write it
divide out any common factor	4x + 6y = 2 is better as 2x + 3y = 1

Worked in full

Find the equation of the line through (2, 1) and (5, 3). Give it in the form ax + by = c, where a, b and c are integers.

gradient = (3 − 1) ÷ (5 − 2) = 2/3

Rise over run, exactly as in section 3.

1 = 2/3 × 2 + c, so 1 = 4/3 + c and c = −1/3

Substitute one of the points, as in the worked examples above.

y = 2/3 x − 1/3

This is the line in y = mx + c form.

× 3: 3y = 2x − 1

Multiply every term by 3 to clear the thirds.

2x − 3y = 1

Move 3y to the right and 1 to the left, so x and y sit together. The x term is positive.

check (5, 3): 2 × 5 − 3 × 3 = 10 − 9 = 1 ✓

Both points must fit. (2, 1): 4 − 3 = 1 ✓

Lines with no y = mx + c form. A vertical line has the same x everywhere: through (4, −1) and (4, 6) it is x = 4. A horizontal line has the same y everywhere: through (−3, 2), parallel to the x-axis, it is y = 2, and its gradient is 0.

The mistake: mixing the two up. A vertical line is x = something, because x is the number that never changes as you go up the line. Say it as you write it: “every point on this line has x equal to 4”.

All yours

Find the equation of the line through (−1, 4) and (3, −2). Give it in the form ax + by = c with whole numbers, like 2x-3y=1.

Write down the equation of the line through (4, −1) and (4, 6).

Write down the equation of the line through (−3, 2) that is parallel to the x-axis.

step 65 · Parallel and perpendicular lines▼

▶ Watch: E3.6 Parallel lines · E3.7 Perpendicular lines

Parallel lines never meet, which means they tilt by exactly the same amount, which means they have exactly the same gradient. That is the whole rule: parallel lines have equal m. The c values differ — that is what stops them being the same line.

y = 2x + 1 and y = 2x − 4 are parallel: same m = 2, different c

Typical question: find the line parallel to y = 3x + 7 through the point (1, 2). Take m = 3 straight from the given line, then find c exactly as in section 4: 2 = 3 × 1 + c, so c = −1, giving y = 3x − 1. Nothing new — the word “parallel” is only there to hand you the gradient.

Perpendicular — and why the rule is what it is

Perpendicular means crossing at a right angle. The rule is that the gradients multiply to −1, so each is the negative reciprocal of the other. That rule is usually handed over as something to memorise. It is worth ninety seconds to see where it comes from, because then you will not misremember it under pressure.

Take a line with run a and rise b, so its gradient is m = ^b⁄_a. Now turn that gradient triangle through a right angle. The side that pointed along now points up, and the side that pointed up now points backwards along the x-axis:

the run a becomes a vertical step of a, pointing down — so it counts as −a;
the rise b becomes a horizontal step of b.

So the turned line has rise −a and run b, and its gradient is −^a⁄_b. Compare the two:

m₁ = ^b⁄_a m₂ = −^a⁄_b → m₁ × m₂ = ^b⁄_a × −^a⁄_b = −1

The a cancels, the b cancels, and all that survives is the minus sign that the turn introduced. That is the entire origin of the rule: flip the fraction, then change the sign.

The sign error to kill here: forgetting that “negative reciprocal” means the sign always changes. If m is already negative, the perpendicular gradient is positive. The perpendicular of −³⁄₄ is +⁴⁄₃, not −⁴⁄₃. Sanity check: two negatives would multiply to a positive, and the product has to be −1, so the two gradients must always have opposite signs. One uphill, one downhill, always.

gradient m	perpendicular gradient	check: product
2	−¹⁄₂	2 × −0.5 = −1
−3	¹⁄₃	−3 × ¹⁄₃ = −1
³⁄₄	−⁴⁄₃	−¹²⁄₁₂ = −1
−¹⁄₅	5	−⁵⁄₅ = −1

Worked in full

Find the equation of the line perpendicular to y = 2x + 1 that passes through (4, 3).

the given line has m = 2

Read it off — the equation is already in y = mx + c form.

flip: ²⁄₁ becomes ¹⁄₂

Writing 2 as 2/1 first makes the flip obvious.

change the sign: perpendicular gradient = −¹⁄₂

Check: 2 × −¹⁄₂ = −1 ✓

y = −¹⁄₂x + c, and (4, 3) is on it: 3 = −¹⁄₂ × 4 + c

Substitute the point, exactly as in section 4.

3 = −2 + c, so c = 5

Half of 4 is 2, and the minus makes it −2. Then add 2 to both sides.

y = −¹⁄₂x + 5

Check: x = 4 gives −2 + 5 = 3 ✓

The perpendicular bisector of a line segment

The perpendicular bisector of AB is the line that cuts AB exactly in half and crosses it at a right angle. Every point on it is the same distance from A as from B. Finding its equation is three steps you already have:

1. midpoint of AB → 2. gradient of AB, then the negative reciprocal → 3. y = mx + c through the midpoint

Worked in full

A is (−3, 8) and B is (9, −2). Find the equation of the perpendicular bisector of AB.

midpoint M = ((−3 + 9) ÷ 2, (8 + (−2)) ÷ 2) = (3, 3)

Average the x values, average the y values. The bisector goes through M.

gradient AB = (−2 − 8) ÷ (9 − (−3)) = −10/12 = −5/6

Rise over run between A and B.

perpendicular gradient = 6/5

Flip 5/6 to 6/5 and change the sign: negative becomes positive. Check: −5/6 × 6/5 = −1 ✓

3 = 6/5 × 3 + c, so 3 = 18/5 + c and c = −3/5

Substitute M, the point the bisector must pass through. Not A, not B.

y = 6/5 x − 3/5, which is y = 1.2x − 0.6, or 6x − 5y = 3

Any of the three forms is the same line.

The mistake: using A or B instead of the midpoint in the last step. The perpendicular bisector passes through the middle of AB, not through its ends. A second slip: keeping the gradient of AB. The word perpendicular means the negative reciprocal.

All yours

Find the equation of the perpendicular bisector of A(−1, 2) and B(5, 6). Write it like y=2x+3.

Find the equation of the perpendicular bisector of P(2, −3) and Q(6, 5). Write it like y=2x+3.

Last step is yours

Find the equation of the line parallel to y = 5x − 2 through the point (2, 4).

parallel → same gradient, so m = 5

Parallel hands you the gradient and nothing else.

4 = 5 × 2 + c = 10 + c, so c = −6

Subtract 10 from both sides. Negative c, and it stays negative.

equation: y = 5x − 6

Check: x = 2 gives 10 − 6 = 4 ✓

Last two are yours

Find the gradient of a line perpendicular to one with gradient −²⁄₅.

flip the fraction: ²⁄₅ becomes ⁵⁄₂

Flip first, deal with the sign separately. Doing both at once is where it goes wrong.

change the sign: the original was negative, so the answer is positive: +⁵⁄₂

The sign always changes. Two negatives could never multiply to −1.

check: −²⁄₅ × ⁵⁄₂ = −¹⁰⁄₁₀ = −1 ✓

Do this multiplication every time. It takes five seconds and it catches the sign error outright.

All yours

A line has gradient −4.

What is the gradient of a line perpendicular to it? Write it like 1/4.

Are y = 3x + 2 and 2y = 6x − 5 parallel? Answer yes or no.

What is the gradient of a line perpendicular to y = −¹⁄₃x + 4?

Find the equation of the line parallel to y = −2x + 9 through (3, 1). Write it like y=-2x+7.

Line A has gradient ⁵⁄₆. Line B is perpendicular to it. What is the gradient of B? Write it like -6/5.

Find the equation of the line perpendicular to y = −¹⁄₂x + 1 through (1, 5). Write it like y=2x+3.

step 5–66 · Length of a line segment, and drawing a line▼

▶ Watch: E3.4 Length and midpoint

The distance between two points is the third side of the same triangle you have been drawing all guide. The run and the rise are the two short sides; the segment itself is the hypotenuse. So it is Pythagoras, nothing more.

length = √( (x₂ − x₁)² + (y₂ − y₁)² )

The one place signs cannot hurt you. Both differences get squared, and squaring destroys a minus sign. So for length — and only for length — it genuinely does not matter which point you subtract first. −3 squared and 3 squared are both 9. Enjoy it; everywhere else in this guide the order matters absolutely.

Worked in full

Find the length of the segment joining (1, 2) and (4, 6).

run = 4 − 1 = 3

The horizontal side of the triangle.

rise = 6 − 2 = 4

The vertical side.

length² = 3² + 4² = 9 + 16 = 25

Pythagoras on the triangle. Square both, then add.

length = √25 = 5

The 3, 4, 5 triangle. Learning 3-4-5 and 5-12-13 by heart saves real time on Paper 2.

Last step is yours

Find the length of the segment joining (−2, −1) and (3, 11).

run = 3 − (−2) = 5

Subtracting a negative adds. The run is 5, not 1.

rise = 11 − (−1) = 12

Same again: 11 + 1 = 12.

length = √(25 + 144) = √169 = 13

The 5, 12, 13 triangle. Worth recognising on sight.

Last two are yours

Find the length of the segment joining (2, 5) and (−4, 13).

run = −4 − 2 = −6, rise = 13 − 5 = 8

The run came out negative, which is fine here — it is about to be squared.

length² = (−6)² + 8² = 36 + 64 = 100

(−6)² = 36, positive. The minus disappears in the squaring.

length = √100 = 10

A length is never negative, so if you ever write one down, something has gone wrong.

All yours

Two points: (−1, 2) and (7, 8).

Find the length of the segment joining them.

Drawing a line from its equation

Two honest methods. Use whichever suits the equation in front of you, but plot three points either way — two define the line and the third catches an arithmetic slip, because three wrong points almost never sit in a straight line.

Method A — a small table. Choose three easy x values, usually 0, 1 and 2 (or −1, 0, 1 if the line is steep), substitute each into the equation, and plot the pairs.
Method B — intercept and step. Mark c on the y-axis. From there, step across 1 and up m, and mark a second point. Repeat. This is faster once you trust it, and it makes the meaning of m visible every time you use it.

Worked in full

Draw y = 3x − 2 using a table.

x = 0 → y = 0 − 2 = −2, giving (0, −2)

x = 0 is always the cheapest point, and it is the y-intercept for free.

x = 1 → y = 3 − 2 = 1, giving (1, 1)

Substitute, do the multiplication first, then the subtraction.

x = 2 → y = 6 − 2 = 4, giving (2, 4)

The third point is the check.

the y values go −2, 1, 4 — rising by 3 each time

Equal steps in x must give equal steps in y, and that step is m. If they are not equal, one point is wrong.

plot the three points and join them with a ruled line right across the grid

Extend the line to the edges of the axes; a short stub between two dots loses marks.

Last step is yours

Find the three points for y = −2x + 5 at x = 0, 1 and 2.

x = 0 → y = 5

The y-intercept.

x = 1 → y = −2 + 5 = 3

Down 2, as the negative gradient promises.

x = 2 → y = −4 + 5 = 1

The y values are 5, 3, 1 — falling by 2 each time, which is exactly m = −2.

Last two are yours

Where does the line y = 4x − 12 cross each axis?

y-axis: put x = 0 → y = −12, so (0, −12)

On the y-axis, x is zero. That is what being on the y-axis means.

x-axis: put y = 0 → 0 = 4x − 12

On the x-axis, y is zero. The two crossings are found by setting the other coordinate to zero.

so 4x = 12, x = 3, giving (3, 0)

Add 12 to both sides, then divide by 4.

All yours

The line y = 5x + 10.

What is the x value where it crosses the x-axis?

Find the length of the segment joining (0, 0) and (9, 12).

Find the length of the segment joining (3, −4) and (3, 6).

What is the value of y on the line y = −3x + 4 when x = 3?

Where does y = 2x − 6 cross the x-axis? Give the x value.

mixed7 · Mixed set — no labels, no order▼

Fifteen questions from all six sections, shuffled and unlabelled. Revision that does one topic at a time quietly does the hardest part for you: deciding which method applies. Here nobody tells you. Before each question, name the type — midpoint, gradient, equation, parallel, perpendicular, length, intercept — and only then start. And on every one that involves a subtraction, predict the sign before you calculate it.

nothing answered yet

1. Find the gradient of the line through (1, −3) and (5, 9).

2. Find the midpoint of (−7, 4) and (3, −10). Write it like (3,4).

3. What is the gradient of a line perpendicular to y = 6x − 1? Write it like -1/6.

4. Find the length of the segment joining (2, 1) and (10, 7).

5. A line has gradient −5 and passes through (1, 2). Find its equation. Write it like y=-5x+7.

6. What is the gradient of 4y = 12x − 20?

7. Find the gradient of the line through (−3, 6) and (5, −2).

8. Where does y = 3x + 9 cross the x-axis? Give the x value.

9. Find the equation of the line parallel to y = −¹⁄₂x + 4 through (2, 1). Write it like y=-0.5x+2.

10. What is the gradient of the line joining (4, 7) and (9, 7)?

11. M(2, −1) is the midpoint of A(−3, 4) and B. Find B. Write it like (3,4).

12. Find the equation of the line through (2, 1) and (5, 10). Write it like y=3x-5.

13. Find the length of the segment joining (−5, 2) and (7, 7).

14. A line is perpendicular to one with gradient ²⁄₇. What is its gradient? Write it like -7/2.

15. Does the point (−2, 11) lie on the line y = −4x + 3? Answer yes or no.

What to take away

Six lines that cover most of what this strand can ask:

Predict the sign before you calculate it. Read the line left to right: rising is positive, falling is negative. Then check your answer against that prediction. This single habit is aimed straight at the error your check found in several strands.
Gradient is rise over run — up on top. Run over rise gets steep lines and gentle ones the wrong way round.
Label your two points and keep the subtractions in the same order on the top and the bottom. Either order works; mixing them is what produces the wrong sign.
Subtracting a negative adds. 3 − (−2) = 5. This is the most common single slip in the whole strand.
y = mx + c only means anything when there is one bare y on the left. Rearrange first, then read m and c off.
Parallel means equal gradients. Perpendicular means flip, then change the sign — so the two gradients always have opposite signs, and their product is −1. Multiply them out to check.

Come back to section 7 in a week without reading anything above it. If the bar sits high on a cold run, step 5 is repaired in this strand.