Linear Equations

6 minQuiz at the end

What is a Linear Equation?

A linear equation produces a straight line when graphed. The standard form is:

y = mx + c

  • m = gradient (slope) β€” how steep the line is
  • c = y-intercept β€” where the line crosses the y-axis

Gradient (Slope)

The gradient tells you how much y changes for each unit increase in x.

m = (yβ‚‚ - y₁) / (xβ‚‚ - x₁)

Example: Through (1, 3) and (3, 7): m = (7-3)/(3-1) = 4/2 = 2

  • Positive gradient β†’ line goes up left to right
  • Negative gradient β†’ line goes down left to right
  • Zero gradient β†’ horizontal line

Graphing a Line

From y = 2x + 1:

  1. Plot the y-intercept: (0, 1)
  2. Use the gradient (up 2, right 1) to find another point: (1, 3)
  3. Draw the line through the points

Checking if a Point is on a Line

Substitute the coordinates into the equation:

  • Does (2, 5) lie on y = 2x + 1? β†’ 2(2)+1 = 5 βœ“ Yes!

Parallel and Perpendicular Lines

  • Parallel lines have the same gradient: y = 3x+1 and y = 3x-4
  • Perpendicular lines have gradients that are negative reciprocals: m₁ Γ— mβ‚‚ = -1