Parallel Lines I
Parallel Lines
Parallel lines have the same gradient.
From this, we get one of the two following cases:
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Two parallel lines that never cross and don't have any points in common.
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Two parallel lines that are identical and share all 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 |
|---|---|---|
| $y=mx+b$y=mx+b | Parallel lines have the same $m$m value. |
$y=2x-1$y=2x−1 $y=4+2x$y=4+2x |
| $ax+by+c=0$ax+by+c=0 | Parallel lines have the same value of $\frac{-a}{b}$−ab. |
$x+2y-3=0$x+2y−3=0 $2x+4y+1=0$2x+4y+1=0 |
For every straight line $y=mx+b$y=mx+b, there exist infinitely many lines parallel to it.
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Here is the line $y=x$y=x
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Here are two more lines in the same family of parallel lines. 
$y=x+1$y=x+1 and $y=x-1$y=x−1 Same gradient ($m$mvalues) Different $y$yintercepts and different $x$x intercepts ($b$b values) |
Examples
Question 1
Is the line $y=4x-1$y=4x−1 parallel to $y=4x-6$y=4x−6 ?
Yes
ANo
B
Question 2
If the line formed by equation $1$1 is parallel to the line formed by equation $2$2, fill in the missing value below.
Equation $1$1: $y$y$=$= $\editable{}$ $x$x$+$+$4$4
Equation $2$2: $y$y$=$=$\frac{4}{5}x$45x$-$−$7$7
Question 3
Consider the following points on the number plane:
$A$A $\left(1,-4\right)$(1,−4)
$B$B $\left(-2,8\right)$(−2,8)
$C$C $\left(-5,2\right)$(−5,2)
$D$D $\left(1,-22\right)$(1,−22)
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 $\left(0,2\right)$(0,2) and $\left(5,27\right)$(5,27) respectively, evaluate the gradient of the line $AB$AB.
If $C$C has the coordinates $\left(1,7\right)$(1,7), evaluate the gradient of the line $BC$BC.
Based on these two gradients, are $A$A, $B$B and $C$C collinear?
Yes
ANo
B