Parallel lines
Changing b, keeping a
Railway lines
Two rails of a railway track run side by side and never meet. Now take the lines y = 2x − 1, y = 2x + 1 and y = 2x + 5. They all have the same a = 2; only b is different.
What do you think the three lines look like together?
What this lesson covers
The idea
Changing b while keeping a fixed shifts the line y = ax + b parallel to itself, so lines with equal slopes but different y-intercepts are parallel.
Railway lines
Two rails of a railway track run side by side and never meet. Now take the lines y = 2x − 1, y = 2x + 1 and y = 2x + 5. They all have the same a = 2; only b is different.
What do you think the three lines look like together?
- They cross each other
- They run side by side, never meeting
- They are the same line
Shift the line
The slope is fixed at a = 2. Change b and press Keep this line. Keep three lines.
Same slope, parallel
If you change b and keep a fixed, the line shifts but stays parallel to the original line. Lines with equal slopes but different y-intercepts are parallel.
y = 2x − 1, y = 2x + 1 and y = 2x + 5 all have slope 2. They are shifted copies of one another and never meet.
Notes
Same slope, different y-intercept: the lines are parallel.
Check yourself
Which pair of lines is parallel?
The lines y = 4x + 2 and y = ax + 7 are parallel. What is a?
Answer: 4
The slopes must be equal, so a = 4.
Two lines have the same slope but different y-intercepts. They are…
The line y = 2x + 1 is shifted 3 units up, parallel to itself. What is the new b?
Answer: 4
The new line is y = 2x + 4, so b = 4.
- y = 3x + 1 and y = x + 1. The slopes are 3 and 1, so these lines are not parallel. They meet at (0, 1).
- y = 2x and y = −2x. The slopes are 2 and −2. They are not equal.
- y = 3x + 1 and y = 3x − 4 — correct. Yes. Both have slope 3 and different y-intercepts.
- parallel — correct. Yes. They never meet.
- the same line. Different y-intercepts mean different lines.
- not related. They are related: each is the other shifted up or down.