Linear Equations
UK Secondary (7-11)
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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=x1

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

    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

    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

    Yes

    A

    No

    B

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