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Algebra · Higher

Straight-line graphs

A straight-line graph has equation y = mx + c, where m is the gradient and c is the y-intercept. At Higher tier you must find equations of lines, work with parallel and perpendicular lines, and use the form ax + by = c.

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Key facts to remember

  • 1Gradient m = (y₂ − y₁) / (x₂ − x₁) — rise over run between any two points.
  • 2y-intercept c is where the line crosses the y-axis (x = 0).
  • 3Parallel lines have the same gradient.
  • 4Perpendicular lines have gradients that multiply to −1 (negative reciprocal).
  • 5To find the equation of a line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁).
  • 6The x-intercept is where y = 0; the y-intercept is where x = 0.
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Formulas

Equation of a line
y = mx + c
Gradient formula
m = (y₂ − y₁) / (x₂ − x₁)
Line through a point
y − y₁ = m(x − x₁)
Perpendicular gradient
m_perp = −1/m
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Worked examples

Example 1

Find the equation of the line through (2, 5) and (4, 11).

Working

  1. Calculate gradient: m = (11 − 5) / (4 − 2) = 6/2 = 3
  2. Use y − y₁ = m(x − x₁) with point (2, 5): y − 5 = 3(x − 2)
  3. Expand: y − 5 = 3x − 6
  4. Rearrange: y = 3x − 1
Answery = 3x − 1
Example 2

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

Working

  1. Gradient of given line is 2.
  2. Perpendicular gradient = −1/2.
  3. Use y − 1 = −½(x − 4).
  4. Expand: y − 1 = −½x + 2.
  5. Rearrange: y = −½x + 3.
Answery = −½x + 3
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Common mistakes

✗Swapping x and y when calculating gradient (doing run/rise instead of rise/run).
✗Thinking parallel lines are perpendicular — parallel lines have equal gradients.
✗Forgetting to use the negative reciprocal for perpendicular gradient.
✗Not rearranging to y = mx + c to read off the gradient and intercept.
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Exam tips

✓Always label your gradient calculation clearly: m = (y₂−y₁)/(x₂−x₁).
✓If the equation is given as ax + by = c, rearrange to y = mx + c first.
✓Perpendicular: flip the fraction and change the sign.
✓Check your answer by substituting a known point into your equation.

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