lesson
Use coordinate change, then recover the intercept
1 · Learn
For two points with different x-coordinates, gradient m=(change in y)/(change in x). Keep the subtraction order consistent in numerator and denominator. Then substitute either point into y=mx+c to find c.
A vertical pair has equal x-coordinates, so division by zero prevents a finite gradient; its equation is x=constant. A horizontal line has gradient zero.
Checking the second point tests both the gradient and intercept. The graph's scale does not change a gradient calculated from coordinate units.
2 · Worked example
Through (1,3) and (4,9), m=6/3=2. Using (1,3): 3=2+c, so y=2x+1. The second point gives 9=8+1.
3 · Your turn
Find the equation through (-1,1) and (3,9).
Check your answer
Gradient 2 and intercept 3, so y=2x+3.
4 · Apply your learning
Plot both points, draw the line, and use a third coordinate from your equation as an independent check.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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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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