Parallel Lines

Parallel lines are lines that have the same gradient
Parallel lines do not intersect with each other
You can easily spot that two lines are parallel when they are written in the form $y = m x + c$ ,
- They will have the same value of $m$ (gradient)
  - $y = 3 x + 7$ and $y = 3 x - 4$ are parallel (and therefore will never intersect)
  - $y = 2 x + 3$ and $y = 3 x + 3$ are not parallel (and therefore must intersect)

If you are given the equations of two lines, rearrange them into the form $y = m x + c$ if they are not in it already
- If the two lines have the same value for $m$ , they are parallel

You may be asked to find the equation of a line parallel to a line of the form $y = m x + c$
- You will also be given some information about a point that the parallel line, $y = m x + d$ passes through; $(x_{1}, y_{1})$
- Substitute into the equation, $y = m x + d$ , the value of $m$ (this will be the same as for the original line) and the coordinates of the point through which it passes $(x_{1}, y_{1})$
- Solve the equation formed to find $d$
- Write the equation of the parallel line substituting in the values of $m$ and $d$

Find the equation of the line that is parallel to $y = 3 x + 7$ and passes through the point (2,1).

As the gradient is the same, the line that is parallel will be in the form

$y = 3 x + d$

Substitute in the coordinates that the line passes through

$1 = 3 (2) + d$

Simplify

$1 = 6 + d$

Subtract 6 from both sides

$- 5 = d$

Write the equation in the form $y = m x + d$ , substituting in the values of $m$ and $d$ that you have found

$y = 3 x - 5$

Parallel Lines (Edexcel GCSE Maths: Foundation)