Hong Kong
Stage 1 - Stage 3

# Parallel Lines I

Lesson

## Parallel Lines

Parallel lines have the same gradient.

From this, we get one of the two following cases:

 Two parallel lines that never cross and don't have any points in common. 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=2x1

$y=4+2x$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+2y3=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.

 Here is the line $y=x$y=x 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=4x1 parallel to $y=4x-6$y=4x6 ?

1. Yes

A

No

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.

1. 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)

1. First, calculate the gradient of the line $AB$AB.

2. Now, find the gradient of the line $CD$CD.

3. Is the line $CD$CD parallel to $AB$AB?

Yes

A

No

B

##### Question 4

Assess whether the points $A$A, $B$B and $C$C are collinear.

1. 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.

2. If $C$C has the coordinates $\left(1,7\right)$(1,7), evaluate the gradient of the line $BC$BC.

3. Based on these two gradients, are $A$A, $B$B and $C$C collinear?

Yes

A

No

B