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Master y = mx + c, gradient calculation, and the parallel/perpendicular conditions. IGCSE worked examples for grades 6-9.
Every straight line has the equation y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). Calculate the gradient using m = (y2 minus y1) / (x2 minus x1) from two known points, then substitute one point into y = mx + c to find c. Parallel lines share the same gradient. Perpendicular lines have gradients that are negative reciprocals (m1 times m2 = minus 1).
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The gradient (or slope) of a straight line measures how steep it is. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line:
m = (y₂ - y₁) / (x₂ - x₁)
A positive gradient means the line slopes upward from left to right. A negative gradient means it slopes downward. A gradient of zero means the line is horizontal. An undefined gradient (division by zero) means the line is vertical.
The order of subtraction does not matter as long as you are consistent — both the y-values and x-values must be subtracted in the same order. Swapping both gives the same result because the two negatives cancel.
The general equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. To find the equation when given two points:
If an equation is given in a different form, you must rearrange to y = mx + c before identifying the gradient. For example, 2y + 4x = 10 rearranges to y = -2x + 5. The gradient is -2, not 4. This trap appears frequently in IGCSE exams and catches many students.
Linear graphs model real relationships. A thermometer converts voltage to temperature using a linear equation: T = mV + c, where m is the conversion rate and c is the offset. Calibrating two reference points (such as ice water and boiling water) determines m and c — exactly the same process as finding the equation of a line from two points.
Two lines are parallel if and only if they have the same gradient: m₁ = m₂. They never intersect. To find a line parallel to a given line through a specific point, use the same gradient and substitute the point to find the new y-intercept.
Two lines are perpendicular (meet at 90 degrees) if the product of their gradients equals -1:
m₁ × m₂ = -1
The mnemonic is "flip the fraction and swap the sign". If one line has gradient 2/3, the perpendicular gradient is -3/2. If one line has gradient -4, the perpendicular gradient is 1/4 (since -4 = -4/1, flipped is -1/4, sign swapped is 1/4).
A perpendicular bisector of a line segment passes through the midpoint at right angles. To find it: calculate the midpoint, find the gradient of the original segment, take the negative reciprocal, then use the midpoint and perpendicular gradient to write the equation. This is a Grade 9 skill that combines multiple concepts.
Find the equation of the line through A(2, 5) and B(6, 13).
Step 1 (Gradient): m = (13 - 5) / (6 - 2) = 8 / 4 = 2
Step 2 (Substitute A into y = mx + c): 5 = 2(2) + c
Step 3 (Solve for c): 5 = 4 + c, so c = 1
Check with B: y = 2(6) + 1 = 13. Correct.
Answer: y = 2x + 1
Find the equation of the line parallel to y = 3x + 7 that passes through (4, 1).
Step 1 (Identify gradient): Parallel means same gradient, so m = 3
Step 2 (Substitute point): 1 = 3(4) + c
Step 3 (Solve for c): 1 = 12 + c, so c = -11
Answer: y = 3x - 11
Find the equation of the perpendicular bisector of the line segment joining P(-2, 4) and Q(8, -2).
Step 1 (Midpoint): M = ((-2+8)/2, (4+(-2))/2) = (3, 1)
Step 2 (Gradient of PQ): m = (-2 - 4) / (8 - (-2)) = -6/10 = -3/5
Step 3 (Perpendicular gradient): Negative reciprocal of -3/5 = 5/3
Step 4 (Substitute midpoint): 1 = (5/3)(3) + c, so 1 = 5 + c, c = -4
Check: At M(3,1): y = (5/3)(3) - 4 = 5 - 4 = 1. Correct.
Answer: y = (5/3)x - 4
Not rearranging to y = mx + c: If the equation is given as 2y + 4x = 10 or 3x - y = 7, you must rearrange before reading the gradient. Reading the x-coefficient directly gives the wrong answer. This is the single most common error in linear graph questions.
Forgetting the negative reciprocal: For perpendicular lines, the gradient is not just the reciprocal — it is the negative reciprocal. If the original gradient is 2/3, the perpendicular gradient is -3/2, not 3/2. Flip AND swap the sign.
Inconsistent subtraction order: In the gradient formula, subtracting y-values in one order and x-values in the reverse order gives the wrong sign. Always subtract in the same order: (y₂ - y₁) / (x₂ - x₁).
Confusing y-intercept with x-intercept: The c in y = mx + c is where the line crosses the y-axis (set x = 0). To find the x-intercept, set y = 0 and solve for x. These are different values.
Rearrange first, always: Before any gradient work, get the equation into y = mx + c form. This one habit eliminates the most common source of errors in linear graph questions.
Verify with the second point: After finding your equation, substitute the other known point. If it satisfies the equation, your answer is correct. This quick check earns confidence and catches arithmetic slips.
State the condition clearly: When a question involves parallel or perpendicular lines, write "parallel means equal gradients" or "perpendicular means m₁ x m₂ = -1" as your first line. This earns the method mark even if the calculation goes wrong.
Sketch a quick diagram: For perpendicular bisector questions, a rough sketch showing the two points, the midpoint, and the direction of the bisector helps you check whether your gradient and intercept are sensible.
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