C: Coordinate geometry in the (x, y) plane: Understand and use the equation of a straight line, including the forms y − y₁ = m(x − x₁) and ax + by + c = 0; gradient conditions for two straight lines to be parallel or perpendicular. Be able to use straight line models in a variety of contexts.

The idea

You need to work confidently with straight lines in different forms. The point-slope form y − y₁ = m(x − x₁) is useful when you know a point and the gradient. The general form ax + by + c = 0 can be rearranged to find gradient and intercepts. For two lines, parallel means equal gradients (m₁ = m₂); perpendicular means gradients multiply to −1 (m₁m₂ = −1), unless one is vertical and the other horizontal. In modelling, straight lines can represent constant rates, such as cost per unit or distance at constant speed. You should interpret gradient and intercepts in context, and use the line to make predictions. Be ready to find equations from graphs, data, or geometric conditions, and to convert between forms.

What earns the marks

• Find the equation of a line given two points or one point and the gradient, using y − y₁ = m(x − x₁).
• Rearrange between y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0 forms.
• Determine whether two lines are parallel, perpendicular, or neither by comparing gradients.
• Find the gradient of a line perpendicular to a given line using the negative reciprocal.
• Interpret the gradient and intercepts of a straight-line model in a real context, including units.
• Use a straight-line model to make predictions or solve problems, and comment on the reasonableness of the model.

Common mistakes and how to put them right

• Forgetting to change the sign when finding the perpendicular gradient.

  Put it right: The perpendicular gradient is the negative reciprocal, so if m = 2, the perpendicular gradient is −1/2.

• Mixing up x and y coordinates when substituting into y − y₁ = m(x − x₁).

  Put it right: Carefully identify (x₁, y₁) and substitute correctly.

• Assuming all straight-line models are valid for all values.

  Put it right: Consider the context and whether the model makes sense for all x, e.g. negative time or distance.

Exam tips

• When given a line in the form ax + by + c = 0, rearrange to y = mx + c to easily identify gradient and y-intercept.
• For perpendicular lines, check that the product of gradients is −1, and remember the special case of vertical and horizontal lines.
• In modelling questions, always relate the gradient and intercepts to the real-world quantities, including units.
• If asked to find a line parallel or perpendicular to a given line through a point, first find the required gradient, then use the point-slope form.