Parallel lines
Identify parallel lines using equal gradients and write equations of lines parallel to a given line.
Worked examples
Identifying parallel lines using gradients
Straightforward
Problem
Consider four lines: , , , and . Which two lines are parallel? Explain why.
1
Find the gradient of each line by comparing with .
: \quad : \quad : \quad :
2
Identify the pair of lines with equal gradients.
and both have gradient . No other pair shares a gradient.
3
Confirm the lines are distinct (different y-intercepts).
The y-intercepts are and , which are different, so the lines are not the same line — they are distinct parallel lines that never intersect.
Answer
and are parallel because they both have gradient .
Writing the equation of a parallel line with a given y-intercept
Moderate
Problem
Line has equation . Write the equation of a line parallel to that has y-intercept .
1
Rearrange the equation of line into form.
2
Read the gradient from the equation.
The gradient of line is .
3
Write the equation of the parallel line using the same gradient and the given y-intercept.
A parallel line has and y-intercept :
4
Check that the lines are distinct and parallel.
Both lines have gradient (parallel). Their y-intercepts are and (different), so they are distinct lines that never meet.
Answer
Writing the equation of a parallel line through a given point
Challenging
Problem
Find the equation of a line parallel to that passes through the point .
1
Identify the gradient of the given line.
The gradient of is .
2
Write the equation of the parallel line in the form .
The parallel line has the same gradient, so its equation is for some unknown .
3
Substitute the given point into the equation to find .
4
Write the final equation.
5
Verify by substituting the point back in.
When : ✓
Answer
Practise
Q1·Straightforward
Which of the following lines is parallel to ?
Explanation
The gradient of is . Checking each option: has ✓; has ; has ; has . Only has the same gradient, so it is the parallel line.
Q2·Straightforward
What is the gradient of any line parallel to ?
Explanation
In , the coefficient of is . Any line parallel to this line must have the same gradient, so its gradient is also .
Q3·Straightforward
Consider the four lines: , , , and . Which pair of lines are parallel to each other?
Explanation
Gradients: has ; has ; has ; has . The pair with equal gradients is and (both have ). They have different y-intercepts ( and ), confirming they are distinct parallel lines.
Q4·Straightforward
What is the gradient of any line parallel to ?
/
Explanation
In , the gradient is . Any line parallel to this has the same gradient: .
Q5·Moderate
A line parallel to has y-intercept . Which of the following is its equation?
Explanation
The gradient of is . A parallel line has the same gradient. Substituting and y-intercept :
The original y-intercept () is replaced by ; the gradient () stays the same.
Q6·Moderate
Line has equation . Line is parallel to line and has y-intercept . What is the equation of line ?
Explanation
Line has gradient . Since line is parallel to , it also has gradient . With y-intercept :
Note: is line itself, not a different parallel line.
Q7·Moderate
A line has equation . A second line is parallel to it and has y-intercept . What is the gradient of the second line?
Explanation
Rearranging :
The gradient is . Any line parallel to this also has gradient .
Q8·Moderate
Which of the following lines is parallel to and has y-intercept ?
Explanation
Rearranging :
The gradient is . A parallel line with y-intercept :
Q9·Challenging
A line is parallel to and passes through the point . Find the y-intercept of this line.
Explanation
The gradient of is . The parallel line has equation . Substituting the point :
The equation of the parallel line is . Verify: ✓
Q10·Challenging
A line is parallel to and passes through the point . What is the y-intercept of this line?
Explanation
Gradient (parallel). Write and substitute :
The equation is . Verify: ✓
Q11·Challenging
A line is parallel to and passes through the point . What is the y-intercept of this line?
Explanation
Gradient (parallel). Write and substitute :
The equation is . Verify: ✓
Q12·Challenging
A line is parallel to and passes through the point . What is the y-intercept of this line?
Explanation
Gradient (parallel to ). Write and substitute :
The equation is . Verify: ✓