Study Notes
The equation of a line describes the relationship between the x and y coordinates for all points on the line. It can be used to determine parallel and perpendicular lines and their gradients.
- Equation of a Line — A mathematical statement that shows how x and y are related. Example: y = mx + c, where m is the gradient and c is the y-intercept.
- Gradient — The steepness of a line, calculated as the change in y over the change in x. Example: For points (x1, y1) and (x2, y2), gradient = (y2 - y1) / (x2 - x1).
- Parallel Lines — Lines that have the same gradient and never meet. Example: y = 2x + 3 and y = 2x - 4 are parallel.
- Perpendicular Lines — Lines that intersect at a right angle, with gradients that multiply to -1. Example: If one line has a gradient of 2, a perpendicular line will have a gradient of -1/2.
Exam Tips
Key Definitions to Remember
- The equation of a line is y = mx + c.
- The gradient (m) is the change in y over the change in x.
- Parallel lines have the same gradient.
- Perpendicular lines have gradients that multiply to -1.
Common Confusions
- Mixing up the x- and y-intercepts.
- Forgetting that perpendicular gradients multiply to -1.
Typical Exam Questions
- What is the equation of a line through points (1, 2) and (3, 12)? Use y = mx + c and calculate m.
- What is the equation of a line parallel to y = 3x + 5? It will have the same gradient, so y = 3x + c.
- What is the equation of a line perpendicular to y = 2x + 3? Use the negative reciprocal of the gradient, so y = -1/2x + c.
What Examiners Usually Test
- Ability to find the equation of a line from two points.
- Understanding of parallel and perpendicular line properties.
- Calculation of gradients and intercepts.