11.16 Parallel lines
Parallel lines
Parallel lines are lines that have the same gradient.
Two parallel lines that never cross and don't have any points in common.
Equations of parallel lines
Let's look at how we can identify parallel lines given their equations.
| Equation Form | Characteristic of parallel lines | Examples |
|---|---|---|
| $y=mx+c$y=mx+c | Parallel lines have the same $m$m value. |
$y=2x-1$y=2x−1 $y=4+2x$y=4+2x |
| $ax+by=d$ax+by=d | Parallel lines have the same value of $\frac{-a}{b}$−ab. |
$x+2y=3$x+2y=3 $2x+4y=-1$2x+4y=−1 |
For every straight line $y=mx+c$y=mx+c, there exist infinitely many lines parallel to it.
Here is the line $y=x$y=x
Here are two more lines in the same family of parallel lines: $y=x+3$y=x+3 and $y=x-6$y=x−6.
Notice that they have the same gradient ($m$m-value) but different $x$x and $y$y-intercepts ($c$c-values).
Lines parallel to the axes
Horizontal lines
Horizontal lines are lines where the $y$y-value is always the same.
Let's look at the coordinates for $A$A, $B$B and $C$C on this line.
$A=\left(-8,4\right)$A=(−8,4)
$B=\left(-2,4\right)$B=(−2,4)
$C=\left(7,4\right)$C=(7,4)
All the $y$y-coordinates are the same, $y=4$y=4.
This means that regardless of the $x$x-value the $y$y value is always $4$4.
The equation of this line is $y=4$y=4
So if the equation of a straight line is $y=c$y=c, then it will be a horizontal line passing through the point $\left(0,c\right)$(0,c).
The $x$x-axis itself is a horizontal line. The equation of the $x$x-axis is $y=0$y=0.
All horizontal lines are parallel to the $x$x-axis and are of the form $y=c$y=c.
They have a gradient of $0$0.
Vertical lines
Vertical lines are lines where the $x$x-value is always the same.
Let's look at the coordinates for $A$A, $B$B and $C$C on this line.
$A=\left(-3,8\right)$A=(−3,8)
$B=\left(-3,3\right)$B=(−3,3)
$C=\left(-3,-3\right)$C=(−3,−3)
All the $x$x-coordinates are the same, $x=-3$x=−3.
This means that regardless of the $y$y-value the $x$x-value is always $-3$−3.
The equation of this line is $x=-3$x=−3
So if an equation of a straight line is $x=c$x=c, then it will be a vertical line passing through the point $\left(c,0\right)$(c,0).
The $y$y-axis itself is a vertical line. The equation of the $y$y-axis is $x=0$x=0.
All vertical lines are parallel to the $y$y-axis and are of the form $x=c$x=c.
Their gradient is undefined.
Practice questions
Question 1
Which lines are parallel to $y=-3x+2$y=−3x+2?
Select the two correct options.
$y=3x$y=3x
A$y=-\frac{2x}{3}+8$y=−2x3+8
B$-3y-x=5$−3y−x=5
C$y=-10-3x$y=−10−3x
D$y+3x=7$y+3x=7
E
Question 2
Write down the equation of a line that is parallel to the $x$x-axis and passes through $\left(-10,2\right)$(−10,2).
Question 3
Consider the following points on the number plane:
$A$A $\left(2,-1\right)$(2,−1)
$B$B $\left(4,-7\right)$(4,−7)
$C$C $\left(-3,1\right)$(−3,1)
$D$D $\left(-6,10\right)$(−6,10)
First, calculate the gradient of the line $AB$AB.
Now, find the gradient of the line $CD$CD.
Is the line $CD$CD parallel to $AB$AB?
Yes
ANo
B
Question 4
Assess whether the points $A$A, $B$B and $C$C are collinear.
If $A$A and $B$B have the coordinates $\left(-4,3\right)$(−4,3) and $\left(-2,7\right)$(−2,7) respectively, evaluate the gradient of $AB$AB.
If $C$C has the coordinates $\left(-7,-3\right)$(−7,−3), evaluate the gradient of $BC$BC.
Based on these two gradients, are $A$A, $B$B and $C$C collinear?
Yes
ANo
B
Question 5
A line goes through A$\left(3,2\right)$(3,2) and B$\left(-2,4\right)$(−2,4):
Find the gradient of the given line.
Find the equation of the line that has a $y$y-intercept of $1$1 and is parallel to the line that goes through $A$A$\left(3,2\right)$(3,2) and $B$B$\left(-2,4\right)$(−2,4).