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Revision guide · IGCSE

IGCSE Maths straight line graphs: gradient, y = mx + c, parallel and perpendicular lines

Every straight line can be written as y = mx + c, where m is the gradient (the change in y for each 1 unit increase in x) and c is where the line crosses the y-axis. To find a line's equation, find m from two points with (y2 - y1)/(x2 - x1), then substitute one point to find c. Parallel lines have equal gradients; perpendicular gradients multiply to -1. Cambridge 0580 Core reads gradients from a grid and uses y = mx + c and parallel lines (C3.1 to C3.6); Extended adds gradients from coordinates, length and midpoint, ax + by = c and perpendicular lines (E3.3 to E3.7). Edexcel 4MA1 Foundation includes midpoints, gradients and y = mx + c (3.3); Higher adds gradients from two points and parallel and perpendicular lines (3.3F, 3.3G).

Facts checked:

At a glance

Cambridge 0580
Core C3.1 to C3.6; Extended E3.1 to E3.7
Edexcel 4MA1
Foundation 3.3; Higher 3.3F, 3.3G
Gradient
(y2 - y1)/(x2 - x1)
Perpendicular
m1 × m2 = -1

What each tier expects

Board and tierReferenceWhat can be asked
Cambridge 0580 CoreC3.1 to C3.6Draw lines from y = mx + c, gradient from a grid only, gradient and intercept from an equation, lines x = k, parallel lines
Cambridge 0580 ExtendedE3.1 to E3.7Gradient from two points, length and midpoint, lines in the form ax + by = c, perpendicular lines and perpendicular bisectors
Edexcel 4MA1 Foundation3.3B to 3.3ICoordinates, midpoint, gradient, y = mx + c, plotting linear graphs including ax + by = c
Edexcel 4MA1 Higher3.3F, 3.3GGradient from two points, equation through two points, parallel and perpendicular lines

The key ideas

The gradient measures steepness: rise divided by run. A line going up from left to right has a positive gradient, one going down has a negative gradient, a horizontal line has gradient 0 and its equation is y = k, and a vertical line has the equation x = k.

To read m and c from an equation, it must be in the form y = mx + c. So rearrange first: 5x + 4y = 8 becomes 4y = -5x + 8, then y = -1.25x + 2, so the gradient is -1.25 and the y-intercept is 2.

To find the equation of a line through two points, calculate the gradient, then substitute either point into y = mx + c to find c. Parallel lines share the same m. For a perpendicular line, take the negative reciprocal: if m = 2, the perpendicular gradient is -1/2.

The midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2), the average of the coordinates. The length of the segment comes from Pythagoras: sqrt((x2 - x1)^2 + (y2 - y1)^2). A perpendicular bisector combines both ideas: it passes through the midpoint at right angles to the segment.

Worked example 1: the line through (1, 7) and (2, 9), and a perpendicular to it

  1. Gradient: m = (9 - 7)/(2 - 1) = 2.
  2. Substitute (1, 7) into y = 2x + c: 7 = 2 + c, so c = 5. The line is y = 2x + 5. Check (2, 9): 4 + 5 = 9. Correct.
  3. A line perpendicular to y = 2x + 5 has gradient -1/2. Find the one through (3, 7): 7 = -1/2 × 3 + c = -1.5 + c, so c = 8.5.
  4. Answer: y = -0.5x + 8.5, which can be written x + 2y = 17. Check (3, 7): 3 + 14 = 17. Correct.

Worked example 2 (Cambridge Extended): perpendicular bisector of (-3, 8) and (9, -2)

  1. Midpoint: ((-3 + 9)/2, (8 + (-2))/2) = (3, 3).
  2. Gradient of the segment: (-2 - 8)/(9 - (-3)) = -10/12 = -5/6.
  3. Perpendicular gradient: the negative reciprocal of -5/6 is 6 over 5, which is 1.2.
  4. Through (3, 3): 3 = 1.2 × 3 + c, so c = -0.6. The perpendicular bisector is y = 1.2x - 0.6, or 6x - 5y = 3 with integer coefficients.
  5. Check: (3, 3) gives 18 - 15 = 3. Correct. The length of the segment is sqrt(12^2 + 10^2) = sqrt(244) = 15.6 to 3 s.f.

Worked example 3: parallel line through a point

  1. Find the equation of the line parallel to y = 4x - 1 that passes through (1, -3).
  2. Parallel means the same gradient, so m = 4 and the line is y = 4x + c.
  3. Substitute (1, -3): -3 = 4 + c, so c = -7.
  4. Answer: y = 4x - 7. Check: 4(1) - 7 = -3. Correct.

Common mistakes that cost marks

  • Subtracting coordinates in a different order on top and bottom, which flips the sign of the gradient.
  • Reading the gradient straight from ax + by = c without rearranging: 5x + 4y = 8 does not have gradient 5.
  • Using the reciprocal without changing the sign for perpendicular lines, or changing the sign without taking the reciprocal.
  • Counting squares on a grid without checking the scale on each axis, so the gradient is out by a factor.
  • Leaving an answer as y - 3 = 1.2(x - 3) when the question asks for y = mx + c or a fully simplified form.
  • Confusing x = 3 (a vertical line) with y = 3 (a horizontal line).

Exam technique and how a tutor helps

Coordinate geometry questions often chain together: midpoint, then gradient, then a perpendicular line, then a point where it meets an axis. Write each result clearly and label it, because later parts use earlier answers and examiners follow through your working. Cambridge asks for equations 'in a fully simplified form', and fractions like 6 over 5 can make decimals the cleaner choice, or multiply through to integer coefficients.

Sketch a quick diagram even when none is given. A rough sketch shows instantly whether your gradient should be positive or negative and whether your midpoint looks sensible.

In one-to-one lessons a tutor can use the shared whiteboard to draw the line as the student calculates, so that the link between the numbers and the picture becomes concrete. For many students the turning point is seeing why perpendicular gradients multiply to -1 by drawing two right-angled triangles, rather than memorising the rule.

Self-check: can you do these?

  • Write down the gradient and y-intercept of y = 3x + 5. (Answer: 3 and 5)
  • Find the gradient of a line perpendicular to 2y = 3x + 1. (Answer: -2/3)
  • Find the midpoint of (2, 5) and (8, -1). (Answer: (5, 2))
  • Find the length of the line from (1, 2) to (4, 6). (Answer: 5)
  • Write the equation of the line with gradient 6 through (0, 2). (Answer: y = 6x + 2)

Common questions

How do I find the gradient from a graph?

Pick two points where the line crosses grid intersections, as far apart as possible. Divide the change in y by the change in x, reading the values from the axes, not by counting squares, in case the scales differ.

What are the gradients of perpendicular lines?

Their product is -1. Take the gradient, flip it and change its sign: 2 becomes -1/2, and -3/4 becomes 4/3. Perpendicular lines are Cambridge Extended and Edexcel Higher content.

Is finding the length between two points on the syllabus?

Yes for Cambridge 0580 Extended (E3.4), using Pythagoras. Edexcel 4MA1 lists the midpoint at Foundation and Pythagoras separately, so the same method answers any length question there.

What does a negative gradient mean in a real-life graph?

That the quantity on the y-axis is decreasing as x increases, for example water draining from a tank. The gradient's size is the rate of change, such as litres per minute.

What does LiveTutor charge for IGCSE Maths?

$15 per 60-minute, one-to-one online lesson, the same rate for every subject and level. The first lesson is a free trial and you can choose 1 to 5 lessons a week, billed monthly.

Sources

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