Linear Graphs

Coordinate Geometry and Graphs

Same gradient, different intercept

Two lines are parallel exactly when their gradients are equal, and they then never meet
y = 7x + 2 and y = 7x − 5 are parallel; y = 7x + 2 and y = 4x + 2 are not
Check by rearranging both equations into y = mx + c first, since a shared gradient can be hidden by the form
To find the equation of a line parallel to a given one, copy the gradient across and then find the new intercept from the point you are given

Exam tip

Parallel means the same gradient and a different intercept. "Parallel to" appears in 11 of the 43 papers, and the usual shape is: take m from the given line, then substitute the point you are given to find c. Copying the whole equation, c included, gives the same line rather than a parallel one.

Worked example

Parallel through a given point

Find the equation of the line parallel to y = 7x + 2 that passes through (3, 8).

Solution:

Parallel means the same gradient, so the new line is y = 7x + c
Substitute x = 3 and y = 8: 8 = 21 + c
Subtract 21: c = −13
The line is y = 7x − 13