- Parallel lines are lines that have the same gradient, but are not the same line
- Parallel lines do not intersect with each other
- You can easily spot that two lines are parallel when they are written in the form , as they will have the same value of (gradient)
- and are parallel (and therefore will never intersect)
- and are not parallel (and therefore must intersect)
- and are not parallel; they are the exact same line

- As parallel lines have the same gradient, a line of the form will be parallel to a line in the form , where is the same for both lines
- If then they would be the same line and therefore not parallel
- If you are asked to find the equation of a line parallel to , you will also be given some information about a point that the parallel line, passes through;
- You can then substitute this point into and solve to find

Find the equation of the line that is parallel to and passes through (2,1)

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

Substitute in the coordinate that the line passes through:

Simplify:

Subtract 6 from both sides:

Final answer:

- You should already know that
**parallel**lines have**equal****gradients** **Perpendicular lines**meet each other at right angles â€“ ie they meet at 90Â°

- Before you start trying to work with perpendicular gradients and lines, make sure you understand how to find the equation of a straight line â€“ that will help you do the sorts of questions you will meet
- Gradients
and*m*_{1}are*m*_{2}**perpendicular**if*m*_{1}Ã—*m*_{2}= âˆ’1- For example
- 1 and âˆ’1
- and 3
- and

- For example
- We can use
*m*_{2}**= âˆ’1 Ã·**to find a perpendicular gradient. This is called the*m*_{1}**negative reciprocal**. -
If in doubt, SKETCH IT!

The line *L* has equation .

Find an equation of the line perpendicular to *L* which passes through the point .

Leave your answer in the form where , and are integers.

*L* is in the form so we can see that its gradient is 2

Therefore the gradient of the line perpendicular to *L* will be the negative reciprocal of 2

Now we need to find for the line we're after. Do this by substituting the point into the equation and solving for

Now we know the line we want is

But this is not in the form asked for in the question. So rearrange into the form where , and are integers

Write the final answer

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