Linear Relationships Math Notes

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Linear Relationships


5ISERIES TOPIC

1Linear RelationshipsMathletics Passport © 3P Learning

Q Twosnailshavespottedthebestlettuceinthegardenandaretravellinginstraightlinestowardsitfordinner.SlidySamisfollowingthepath: 2y x2= - andSlipperyShellyisfollowingthepath:2 2y x= - - Whichlettucearetheybothmovingtowardsifitisatthepointwheretheirpathscross?

Work through the book for a great way to solve this

Give this a go!

Writeasinglesentencethatdescribesthemathematicalmeaningfortheseterms:Youcanuseasmalldiagram/exampletosupportyourdescription.

Relationship LinearContinuous

Thisbookletextendsonstraightlineconceptsbyinvestigatingtherelationshipbetweenpointsonaline

Jointhesedefinitionstogethertodescribeinaonesentencewhatyouthinkismeantbyalinear relationship.

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y-axis

1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0

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A BC

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LettuceSamandShellyaremovingtowards:

2 Linear RelationshipsMathletics Passport © 3P Learning

5ISERIES TOPIC

Linear RelationshipsHow does it work?

Fortheequationy x3 4= -

Gradient-intercept form

Alinearrelationshipisanalgebraicrulethatformsacontinuous,straightlinewhengraphed.Let’sreviewthebasiclinear equation.

(i) Writethegradientandy-interceptforthegraphofy x3 4= -

gradient m 313= =

++^ h y-intercept b 4= -^ h

(ii) Describewhatthesevaluesmeanforthegraphofy x3 4= -

y mx b= +

coefficientofx (m)isthegradient theconstantterm(b)isthey-intercept

Beforegraphingitisimportanttobeabletoextractthecorrectinformationfromalinearequation.

thepowerofxandyisalways1

gradient-interceptform

y x3 4= -

:m 3= graphmoves 3+ vertically(slopesup)forevery 1+ horizontally

:b 4= - graphpassesthroughthepoint ,0 4-^ hor

thegraphcrossesthey-axisat 4-

Fortheequationy x 2= +

(i) Writethegradientandy-interceptforthegraphofy x 2= +

gradient 1m11= =

++^ h y-intercept b 2= +^ h

The 1infrontofthex ishidden

(ii) Describewhatthesevaluesmeanforthegraphofy x 2= +

Sometimestheinformationneededcanappeartobe‘hidden’.

:m 1= graphmoves 1+ vertically(slopesup)forevery 1+ horizontally

:b 2= + graphpassesthroughthepoint ,0 2^ hor

thegraphcrossesthey-axisat2

y x 2= +

Equationisanother wordforrule

5ISERIES TOPIC


How does it work? Linear Relationships

y x31

21= - +y x

3 21= - +

y x32 1= +

Fortheequationy x32 1= +

Herearesomemoreexampleswithdifferentpointstolookoutfor.

(i) Writethegradientandy-interceptforthegraphofy x32 1= +

gradient m32

32= =


(ii) Describewhatthesevaluesmeanforthegraphofy x32 1= +

:m32= graphmoves 2+ vertically(slopesup)forevery 3+ horizontally

:b 1= + graphpassesthroughthepoint ,0 1^ h or thegraphcrossesthey-axisat 1

Fortheequationy x4 5= - +

(i) Writethegradientandy-interceptforthegraphofy x4 5= - +

gradient m 414= - =

+-^ h y-intercept b 5= +^ h

(ii) Describewhatthesevaluesmeanforthegraphofy x4 5= - +

Becarefultocorrectlyinterpretthemeaningofnegativesigns.

:m 4= graphmoves 4- vertically(slopesdown)forevery 1+ horizontally

:b 5= + graphpassesthroughthepoint ,0 5^ h orthegraphcrossesthey-axisat5

y x4 5= - +

Writethegradientandy-interceptforthegraphofy x3 2

1= - +

Whenthecoefficientofxisafraction,itcanappearinaslightlydifferentform.

gradient m31

31= - =

+-^ h y-intercept b

21=^ h

Writethegradientandy-interceptforthegraphof 3 2y x= -

Don’tletexpressionswritteninadifferentordertrickyou.

y-intercept b 3= +^ h gradient m 212= - =

+-^ h

3 2y x= -

or

Remember:Thecoefficientisthenumberinfrontofthevariable

sameas 2 3y x= - +


5ISERIES TOPIC

How does it work? Linear RelationshipsYour Turn


1 (i) Writethegradientandy-interceptforeachoftheselinearequations(ii) Describewhatthesevaluesmeanforthegraphofeachlinearequation

GRADIENT-INTERCEPT FORM * GRADIENT-IN

TERCEPT FORM

*

..../...../20...

a y x2 1= - b 6y x= -

Gradient(m) =

y-intercept(b)=

(i)

(ii) Thegraphofy x2 1= - moves:

verticallyforeveryhorizontally

Gradient(m) =

y-intercept(b)=

(i)

(ii) Thegraphofy x 6= - moves:

c y x3 2= - + d y x5 2= +

Gradient(m) =

y-intercept(b)=

(i) Gradient(m) =

y-intercept(b)=

(i)

e y x 6= - + f y x341= - +

Gradient(m) =

y-intercept(b)=

(i) Gradient(m) =

y-intercept(b)=

(i)


(ii) Thegraphofy x3 2= - + moves:


(ii) Thegraphofy x5 2= + moves:


(ii) Thegraphofy x 6= - + moves:


(ii) Thegraphofy x341= - + moves:


y mx b=

+

5ISERIES TOPIC




1 (i) Writethegradientandy-interceptforeachoftheselinearequations(ii) Describewhatthesevaluesmeanforthegraphofeachlinearequation

a 2y x= + b y x21

32= +

Gradient(m) =

y-intercept(b)=

(i)

(ii) Thegraphof 2y x= + moves:


Gradient(m) =

y-intercept(b)=

(i)

(ii) Thegraphofy x21

32= + moves:

c 3y x= - d y x2

8= - +

Gradient(m) =

y-intercept(b)=

(i) Gradient(m) =

y-intercept(b)=

(i)

e y x31= - - f y x

54

23= -

Gradient(m) =

y-intercept(b)=

(i) Gradient(m) =

y-intercept(b)=

(i)


(ii) Thegraphof 3y x= - moves:


(ii) Thegraphofy x2

8= - + moves:


(ii) Thegraphofy x31= - - moves:


(ii) Thegraphofy x54

23= - moves:



5ISERIES TOPIC


Writethegradientandy-interceptforthegraphof2 6 12y x- =

Rearranging linear relationships

Somelinearrelationshipsneedtobere-arrangedbeforethegradientandy-interceptcanberead.

move6xovertotherighthandside

gradient m 313= =


Usingasimilarmethodtosolvingequations,makeythesubjectoftheformula.

2 6 12y x- =

3 6y x= +

2 6 12y x= +

x x6 6+ +

2 2 2' ' '

Writethegradientandy-interceptforthegraphof2 3 9x y+ =

gradient m32

32= - =


Leavevaluesassimplifiedfractionsafterrearranging.

y x

y x

332

32 3

= -

= - +

3 9 2y x= -

x x2 2- -

3 3 3' ' '

Writethegradientandy-interceptforthegraphof 2y x9

3 -=

gradient m31

31= =


Hereisanotherexamplewherea1ishiddenattheend.

6y x3

= +

1y x3 8= +

x x+ +

2 3 9x y+ =

divideeverytermby2toisolatey

linearequationisnowintheform y mx b= +




9 2 9y x9

3##

-=

1y x3 8- =

3 3 3' ' '

multiplybothsideby9

movexovertotherighthandside


5ISERIES TOPIC




1 (i) Rearrangeeachoftheselinearrelationshipsintogradient-interceptform y mx b= +^ h(ii) Writethegradientandy-interceptforeachoftheselinearequations

a 4 8 12y x= - b 2 14 6y x= +

Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(i)

(ii)

c 10 10 25y x- = d 4 3 12y x+ =

e 6 2 1x y+ = f 8 4 16x y- =

Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(ii)

Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(i)

(ii)

hint:becarefulwithnegativevalueshere

..../...../20...REARRA

NGING LINEAR RELATI

ONSHIPS * REARRANGING LINEAR RELATIONSHIPS *

ymx b

=+


5ISERIES TOPIC



2 (i) Rearrangeeachoftheselinearrelationshipsintogradient-interceptform y mx b= +^ h(ii) Writethegradientandy-interceptforeachoftheselinearequations

a 1y

x3

= - b 2 3y x21 = +

Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(i)

(ii)

c 2y x3-

= d 6y x5

2 6+=

e 1y x2

5 4+= f 5 3y x

35+ =

Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(i)

(ii)

Gradient(m) =

y-intercept(b)=

(i)

(ii) Gradient(m) =

y-intercept(b)=

(i)

(ii)

hint:becarefulwiththesignofthegradient

5ISERIES TOPIC



Fortheequation 2y x= +

Graphing using the intercept and gradient

Afterfirstplottingthey-intercept,thegradientcanthenbeusedtofindasecondpointtohelpdrawthegraph.

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Step 1:Plotthey-intercept

1 2 3 4 5 -5 -4 -3 -2 -1

(i) Writethegradientandy-interceptforthegraphof 2y x= +

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6

5

4

3

2

1

0

-1

-2

-3

Step 3:Ruleadoublearrowedline throughthepoints

1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

y-axis

x-axis

y-axis

gradient m 111= =


` Graphoftherulemoves 1+ vertically(slopesup)forevery 1+ horizontallyandpassesthroughthey-axisat(0 , 2)

(ii) Describewhatthesevaluesmeanforthegraphofy x 2= +

1+

Step 2:Usethegradienttoplotasecondpoint

Step 4:Writetherulealongtheline

1+

2y x= +y

x2

=

+


5ISERIES TOPIC


Fortheequation y x3 2 12+ =

Rearrangelinearrelationshipsfirsttofindthey-interceptandgradient.

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6

5

4

3

2

1

0

-1

-2

-3

Step 1:Plotthey-intercept

1 2 3 4 5 -5 -4 -3 -2 -1

(i) Writethegradientandy-interceptforthegraphof y x3 2 12+ =

7

6

5

4

3

2

1

0

-1

-2

-3

Step 3:Ruleadoublearrowedline throughthepoints

1 2 3 4 5 6 7 -5 -4 -3 -2 -1 x-axis

y-axis

x-axis

y-axis

(ii) Usethisinformationtographthelinearequation y x3 2 12+ =

2-

Step 2:Usethegradienttoplotasecondpoint

Step 4:Writetherulealongtheline

3+


gradient m32

32= - =


y x

y x

432

32 4

= -

= - +

y x3 12 2= -

x x2 2- -

3 3 3' ' 'divideeverytermby3toisolatey


y x3 2 12+ =

yx

32

12

+

=

Rememberthegradientis:

Howfarup(+)ordown(-)

Howfaracross(+)

5ISERIES TOPIC




1 Sketcheachoftheselinearequationsusingthey-interceptandgradient

a y x2 3= + b 3 1y x= - +

Gradient(m) =

y-intercept(b)=

Gradient(m) =

y-intercept(b)=

c y x21 3= + d y x

52 2= +

GRAPHING USING THE INTERCEPT A

ND GRADIENT

*

..../...../20...

Gradient(m) =

y-intercept(b)=

Gradient(m) =

y-intercept(b)=

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

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6

5

4

3

2

1

x-axis

y-axis

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

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6

5

4

3

2

1

x-axis

y-axis

0

-1

-2

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1 2 3 4 5 -5 -4 -3 -2 -1

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4

3

2

1

x-axis

y-axis

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

7

6

5

4

3

2

1

x-axis

y-axis


5ISERIES TOPIC



2 Sketcheachoftheselinearequationsusingthey-interceptandgradient

a 2y x 4= - - b y x 5= - -

Gradient(m) =

y-intercept(b)=

Gradient(m) =

y-intercept(b)=

c 3y x53= - d y x

35= -

Gradient(m) =

y-intercept(b)=

Gradient(m) =

y-intercept(b)=

7

6

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

y-axis7

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5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

y-axis

7

6

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

y-axis7

6

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

y-axis

5ISERIES TOPIC




3 (i) Writethegradientandy-interceptforeachoftheselinearequations(ii) Usethisinformationtographeachlinearequation

a y x4 3 4- = -

Gradient(m) =

y-intercept(b)=

(i) (ii)

b y x3

3 21

+=

(i) (ii)

c x y2 329+ =

(i) (ii)

Gradient(m) =

y-intercept(b)=

Gradient(m) =

y-intercept(b)=

7 6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

y-axis

0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

7 6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

y-axis

0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

7 6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

y-axis

0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis


5ISERIES TOPIC


The linear equation from the graph

Byworkinginreversetobefore,wecanfindthey-interceptandgradientoftheline.Thesevaluesarethensubstitutedintothegradient-interceptformula.

Findtheequationofthelineforthelineargraphbelow:

Findtheequationofthelineforthislineargraph

Forgraphswithonlyonevisibleintercept,searchforanothereasy-to-readpoint.

x-axis

y-axis7

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4

3

2

1

0

-1

-2

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Step 1:Readthey-intercept

1 2 3 4 5 -5 -4 -3 -2 -1

x-axis

y-axis7

6

5

4

3

2

1

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

2+

Step 2:Determinethegradientinsimplestform

y mx b= +

y-intercept b 0=^ h Gradient m53

53=

+- = -^ h

y-intercept b 2= +^ h Gradient m32

32=

++ =^ h

` y x53= -

y mx b= +

y-intercept 0=

5+

3+

2y x32` = +

3-

5ISERIES TOPIC




Matchtheletterofeachgraphwithitslinearrelationship.

A

x-axis

y-axis

1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1

-1

-2

-3

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1

0

N P

G

C

H

S

I C R

E

I

5 7 35x y+ =-5x= y x8

4 8= +2 3y x= +3 8 0x y+ =y x=1y

x2= -4 20y x= -

y x3 1= - y x6+ =y x7= +2y x= THE

LINEAR EQUAT

ION FROM THE GRAPH

..../...../20...


5ISERIES TOPIC


Comparing graphs

Bygraphingmorethanonelinearequationontothesamesetofaxes,wecanseeifthereisaspecialrelationshipbetweenthem.

Let’sseewhathappenswhentwolinearequationswiththesamegradientareplottedtogether.

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2

1

0

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-2

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Allthreegraphsgothroughthesamepoint:(2 , 2)

` ifyousubstitutex = 2intoanyoftheseequations,ywillbe2

1 2 3 4 5 -5 -4 -3 -2 -1

For:y 2=

For:y x21 1= +

For:y x4= -

Graphtheseequationsonthesamesetofaxes: , y x221 1= = + andy x4= -

Samepoint =

Concurrentlines

m 0= andb 2= +

1m2

= andb 1= +

x-axis

y-axis7

6

5

4

3

2

1

0

Parallellinesnevercrosseachother

1 2 3 4 5 -5 -4 -3 -2 -1

For:y x 1= +

For:y x 3= -

Graphthelinearequations 1y x= + andy x 3= - onthesamesetofaxes

Samegradient =

Parallellines

m 1= andb 1= +

m 1= andb 3= -

-1

-2

-3

-4

-5

1

y

x=

+

y

x3

=

-

Theyallcrosshere

1m = - andb 4= +

Let’sseewhathappenswhenthesethreeequationsaregraphedonthesamesetofaxes.

Where does it work?

y

x4

=

-

(horizontalline)

.1

y

x0 5

=

+

y 2=

5ISERIES TOPIC


Where does it work? Linear RelationshipsYour Turn

Comparing graphs

1 Comparethegradientsforeachofthefollowingpairsoflinearequations andtickwhethertheyareparallelornot parallel.

a y x2 3= + and 1 2y x= + b x y 4+ = andy x 4= -

Parallel

c y x3

= andy x131= + d y x2 2- = andy x2 2= -

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

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1

y-axis

2 Ineachofthese,onelinearequationhasalreadybeengraphedbutnotlabelled. Comparethegradientofthegraphedequationwiththeun-graphedequation. Tickwhethertheywouldbeparallelornot parallelwhengraphedonthesamesetofaxes.

psst!:Youcangraphtheotherequationifithelps!

a y x 4= +

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

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5

4

3

2

1

y-axis

NotParallel Parallel NotParallel

Parallel NotParallel Parallel NotParallel

Remember:Parallellineshavethesamegradient

Parallel NotParallel

x-axis

COMPAR

ING GR

APHSCOMPARING GRAPHSCOMPARING GRAPHSCO

MPARING GRAPHS

..../...../20...


5ISERIES TOPIC


Comparing graphs

b y x32 2= -


c y x3 9= Parallel NotParallel

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

y-axisd y x2 2 3+ = -

7

6

5

4

3

2

1


x-axis

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

y-axis

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6

5

4

3

2

1

x-axis

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

y-axis

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6

5

4

3

2

1

x-axis

5ISERIES TOPIC



a , 4 , 4 2 4y x x x= = = -

Concurrent NotConcurrent

b , ,y x y y x21 1 1= = = -

Comparing graphs

3 (i) Sketcheachofthethreelinearequationsonthesamesetofaxes(ii) Tickwhethertheywouldbeconcurrentornot concurrent

Remember:Concurrentlinesallpassthroughthesamepoint


c 3 6 , 2 , 2 6y x x y x6 8+ = = + =


7 6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

y-axis

0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

7 6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

y-axis

0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis

7 6 5 4 3 2 1

-1 -2 -3 -4 -5 -6

y-axis

0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis


5ISERIES TOPIC

Linear RelationshipsWhat else can you do?

Intersection of two linear graphs

Wherethegraphsoftwolinearrelationshipscrosseachotheriscalledtheirintersection point. Atthispoint,thesamexandyvaluesworkinbothlinearequations.

x-axis

y-axis7

6

5

4

3

2

1

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

For: 2 1y x= +

For: 4y x= -

Fortheequations 1y x2= + andy x4= -

m 1= andb 1= +

1m = - andb 4= +

y

x4

=

-

1

yx2

=+

(i) Graphthelines 2 1y x= + and 4y x= - onthesamenumberplane

Drawverticalandhorizontallinesfromtheintersectionpointtotheaxestofindthecoordinates(ii) Writedowntheirpointofintersection

x-axis

y-axis7

6

5

4

3

2

1

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

y

x4

=

-

1

yx2

=+

Pointofintersection (1 , 3)

Writedownthepointofintersectionforthelinesx 4= andy 1=

Theintersectionofhorizontalandverticalgraphsiseasilyfoundbycombiningtheirequations.

x 4= y 1=

` Pointofintersectionissimply:(4 , 1)

5ISERIES TOPIC


What else can you do? Linear Relationships

Hereisasimilarexampleinadifferentcontext.

Twotreasurehuntersarewalkinginstraightlinestowardhiddentreasurealongtwodifferentpaths. Theirpathscrossattheexactlocationofthetreasure.

(i) “One-eyedPi”iswalkingalong2 4y x= +

“Two-eyedSquinty”iswalkingalong 6x =

Graphtheirpathstodeterminethelocationofthetreasure

Thesegraphsintersecteachotherat (6 , 5)

y x2

2` = +

2m b21` = =

` thetreasureisatcoordinates(6 ,5)

(ii) Athirdtreasurehunterbelievesthetreasureliesonthepath 3y x= - .Ifheiswalkingfromthethirdquadrantalongthispath,showwhetherhewillnotfindthetreasureornot.

One-eyedPi:2 4y x= + Two-eyedSquinty: 6x =

` when ,x y6 6 3= = -

= 3

Treasureisat(6 , 5)

` whenx 6= ,hewillbeatcoordinates(6 , 3), 2 unitsawayfromthetreasure ` hewillnotfindthetreasureonthispath.

x-axis

y-axis

1 2 3 4 5 6 7 8

-1

-2

-3

-4

-5

-6

-7

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2 -1 0

substitute 6x = intotheequation

6x=

2

4

yx

=+


5ISERIES TOPIC

What else can you do? Linear RelationshipsYour Turn

a y x3= - andy x2= b y x2 2= - andy x 4= - +

Pointofintersection: ( , )

c y 3= andx 1= - d x 3= - andy x2 1= - -


1 (i) Grapheachpairofequationsbelowonthesamenumberplane(ii) Writethecoordinatesoftheirpointofintersection


Pointofintersection: ( , ) Pointofintersection: ( , )

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

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1

x-axis

y-axis

0

-1

-2

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1 2 3 4 5 -5 -4 -3 -2 -1

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2

1

x-axis

y-axis

0

-1

-2

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1 2 3 4 5 -5 -4 -3 -2 -1

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2

1

x-axis

y-axis

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

7

6

5

4

3

2

1

x-axis

y-axis

INT

ERSECTION

OF TWO LINEAR GRAPHS * * *

..../...../20...

5ISERIES TOPIC



a y x 2= + andy x53= b 2y x

21= + andy x

23 6= +


c y x 2= - andy x42

= - d y x= - and 2 1y x4 1 6= +


2 (i) Grapheachpairofequationsbelowonthesamenumberplane(ii) Writethecoordinatesoftheirpointofintersection


0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

7

6

5

4

3

2

1

x-axis

y-axis

Pointofintersection: ( , ) Pointofintersection: ( , )

0

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1 2 3 4 5 -5 -4 -3 -2 -1

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5

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3

2

1

x-axis

y-axis

0

-1

-2

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1 2 3 4 5 -5 -4 -3 -2 -1

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1

x-axis

y-axis

0

-1

-2

-3

1 2 3 4 5 -5 -4 -3 -2 -1

7

6

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1

x-axis

y-axis


5ISERIES TOPIC


Remember me?


3 Twosnailshavespottedthebestlettuceinthegardenandaretravellinginstraightlinestowardsit.

*

AWESOM

E *

..../...../20...

* AWESOME *

(i) Graphthepathofeachsnail.

(ii) Whichlettucearetheymovingtowardsifitisatthepointwheretheirpathscross?

SamandShellyaremovingtowardslettuce:

(iii) SluggySteveisalsointhegardenslidingtowardsalettucealongthepath2 3 12y x= - . ShowthatSluggySteveisnotmovingtowardsthesamelettuceasthetwosnailsbygraphinghispath. WhichlettuceisSluggyStevemovingtowards?

SluggySteveismovingtowardslettuce:

(iv)Writedownthelinearequationsfortwodifferentpathsthatpassthroughthecentreofthreelettuces. hint:findthepossiblepathsfirstusingaruler

Firstpathpassingthroughthecentreofthreelettuces:

Secondpathpassingthroughthecentreofthreelettuces:

SlidySamisfollowingthepathy x2 2= -

SlipperyShellyisfollowingthepath y x2 2= - -

x-axis

y-axis

1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0

-1

-2

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-6

-7

7

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A BC

E

I

J

H

F

D

G

5ISERIES TOPIC




4 DaniandReecebothrodetheirbicyclesthroughaparktwiceatdifferenttimesoftheday.

(i) Graphthepathofeachridethroughthepark.

(ii) HowmanytimesdidDani’sridingpathscrosswithReece’swithinthepark?

DaniandReececrossedpaths timeswithinthepark.

(iii)WritedownallthecoordinateswhereDani’spathscrossedwithReece’s.

(iv)Writetwoequationsfordifferentpathsthroughtheparkthat only crossthroughoneofReece’spaths.

Firstpaththrough:

Secondpaththrough:

Danirodealongthefollowingpaths: 2 6y x= - and 3y x+ = -

Reecerodealongthefollowingpaths:6 12y x= - and3 9 21y x= -

x-axis

y-axis

-8 -7 -6 -5 -4 -3 -2 -1 0

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-7

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1

1 2 3 4 5 6 7 8


5ISERIES TOPIC


Reflection Time

Reflectingontheworkcoveredwithinthisbooklet:

1 Whatusefulskillshaveyougainedbylearningaboutlinearrelationships?

2 Writeaboutoneortwowaysyouthinkyoucouldapplylinearrelationshipstoareallifesituation.

Ifyoudiscoveredorlearntaboutanyshortcutstohelpwithlinearrelationshipsorsomeothercoolfacts,jotthemdownhere:

3

5ISERIES TOPIC


Cheat Sheet Linear Relationships

Here is a summary of the important things to remember for linear relationships



Afterfirstplottingthey-intercept,thegradientcanthenbeusedtofindasecondpointtohelpdrawthegraph.


Byworkinginreversetobefore,youcanfindthey-interceptanddeterminethegradientoftheline.Thesevaluesarethensubstitutedintothegradient-interceptformula.

Comparing graphs

• Graphswiththesamegradientareparalleltoeachother(theynevercross) • Ifanumberofgraphspassthroughthesamepoint,thesegraphsarecalledconcurrent graphs


Wherethegraphsoftwolinearrelationshipscrosseachotheriscalledtheirintersection point. Atthispoint,thesamexandyvaluesworkinbothlinearequations.

Picture Summary

y mx b= +

theconstantterm(b)isthey-intercept

thepowerofxandyisalways1

gradient-interceptform

coefficientofx (m)isthegradient

x-axis

y-axis

7

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2

1

0

Parallellines(samegradient)

1 2 3 4 5 -5 -4 -3 -2 -1

-1

-2

-3

-4

-5

-6

y

x3

22

=

+

Concurrentlines(allintersectatthesamepoint)

Intersectionpoints(wherelinescross)

Negativegradient(slopingdown)

Positivegradient(slopingup)

y-intercept

x-intercept

2

y

x3

2=

-


5ISERIES TOPIC

Linear Relationships Notes

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Linear Relationships Math Notes

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