lesson
Equal gradients do not mean the same line
1 · Learn
Lines y = mx + c with equal gradients are parallel unless their intercepts also agree, in which case they are the same line. To find a line through a point, substitute that point to determine the missing intercept.
Keep coordinate order correct and check a second point when available. A vertical line does not have a finite gradient in this y = mx + c form.
2 · Worked example
A line parallel to y = 2x + 1 through (3,8) has form y = 2x + c. Since 8 = 6 + c, c = 2.
3 · Your turn
Find the line of gradient 3 through (2,7).
Check your answer
y = 3x + 1.
4 · Apply your learning
Plot two parallel examples and explain why their constant vertical separation is not their gradient.
Adult guidance and safety
Check prerequisites, units and reasoning. Provide accurate diagrams and varied practice. Extension content needs suitable prior understanding; no GCSE board, tier or examination readiness is implied by this proposed sequence.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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