straight line

Straight line in real-life

Equation given any two points

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Gradient Revision

Nat 5

The General Equation of a straight line.

Best – fitting straight line

Straight LineStraight Line

Simple Gradient

Equation of a line gradient and y-intercept

Exam Type Questions

Higher Form of straight Line y – b = m(x – a)

20 Apr 202320 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept

Starter QuestionsStarter Questionsw

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Nat 5

In pairs

“Explain the rules for fractions”

20 Apr 202320 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww

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2.2. Calculate simple Calculate simple gradients.gradients.

1. We are learning to calculate simple gradient using a right angle triangle

1.1. Gradient is : Gradient is : change in vertical height change in vertical height

divided by divided by change in horizontal change in horizontal

distancedistance

The Gradient of a The Gradient of a LineLine

S4


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The GradientThe Gradient

Change in vertical height

Change in horizontal distance

The gradient is the measure of steepness of a line

V

H

The steeper a line the bigger the gradientDifference in x -

coordinates

Difference in y -

coordinatesS4


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The GradientThe Gradient

0 2 4 6

2

4

6

-6

- 4

-2

-2- 4-6

V

mH

3

4

V

mH

3

2

V

mH 5

V

mH

2

6

1

3

3

S4


Now try N5 TJ Ex 6.1

Ch6 (page 50)

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Straight LineStraight LineNat 5



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Nat 5

98

5

Q1. Is this triangle right angled ?Explain


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1. We are learning to find the equation of a straight line.

1.1. Understand how to Understand how to calculate the gradient and calculate the gradient and identify y-intercept.identify y-intercept.

Straight lineStraight line

Nat 5

2.2. Be able to write down Be able to write down equation of a straight line equation of a straight line in the format y = mx + c.in the format y = mx + c.


Almost all straight lines have the equation of the form

y = mx + c

Gradienty - intercept

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Straight Line Straight Line EquationEquation

y intercept is were line

cuts y axis

Slope left to right upwards

positive gradient

Slope left to right

downwards negative gradient

Nat 5

lines are parallel if

same gradient

mm = gradient = gradient

x1

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y-axis2 2(x ,y )

2 1

2 1

y yx

mx

2 1y y

O x-axis

2 1x x

1 1(x ,y )

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The gradient The gradient using coordinatesusing coordinates

x2

y2

y1

Nat 5


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2 1

2 1

= m

y ygradient

x x

It is a measure of how steep a line is

A line sloping up from left to right is a positive gradient

A line sloping down from left to right is a negative gradient


Nat 5

Remember parallel means same gradient

2 1

2 1

y yx

mx

8 2

3 ( 1)m

6

32

m


y-axis

(3,8) 2 1

2 1

y yx

mx

O x-axis

(1,2)


Find the gradient of the two lines.

8 2

3 1m

6

32

m

(-3,8)

(-1,2)

Nat 5



Write down the gradient and y intercept for each line.

Nat 5

(a) y = -3x - 5

(b) 4y - 8x = 24

m = - 3 c = - 5

m = 2 c = 6

Write the equation given the gradient and y intercept.

(a) m = 1.5 c = 1

(b) m = -2 c = - 4

y = 1.5x + 1

y = -2x - 4

Challenge

http://ggbtu.be/mlpi4kd7n

http://ggbtu.be/mlpi4kd7n



Ch6 (page 52)

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Calculate it’s value if it has increased by 5%.

Q2. Calculate 3.36 x 70 to 2 significant figures.

Q3. Write down the 3 ways of factorising.

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2.2. Know how to find y-Know how to find y-intercept using intercept using substitution . Be able to substitution . Be able to write down straight line write down straight line equation in y = mx + c equation in y = mx + c format.format.

1. We are learning to find the equation of a straight line given any two points.

1.1. Know how to calculate the Know how to calculate the gradient using formula.gradient using formula.

Straight LineStraight Line

Nat 5

Apr 20, 2023Apr 20, 2023 www.mathsrevision.comwww.mathsrevision.com

Equation of a Straight Line

To find the equation of a straight line we need to know

Two points that lie on the line

( x1, y1) and ( x2, y2)

OR

The gradient and a point on the line

m and (a,b)

y = mx + c

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Nat 5

We will look at this later

20 Apr 2023

-10

y

Find the equations of the straight lines below given the coordinates.

2 1

2 1

y ym

x x

(4,1) and (8,5)

using

5 1m

8 4

Sub (4,1) intoy = mx + c

x y

1 = 1x4 + cc = 1 - 4c = -3

y = x -3

41

4

2 1

2 1

y ym

x x

(-6,7) and (-2,3)

3 7m

( 2) ( 6)

Sub (-2,3) into y = mx + c

x y

c = 3 - 2c = 1

y = -x + 1

41

4

3 ( 1) ( 2) c

Demo

http://ggbtu.be/mDkqTEQ10

Find the equation of the straight line passing through the points (4, 4) and (8,24).

Solution2 1

2 1

y y 24 4 20m 5

x x 8 4 4

y = mx + c

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Nat 5


Using the point (4,4) and the gradient m = 5 sub into straight line equation

4 = 5 x 4 + c c = 4 - 20 = -16

Equation :y = 5x - 16

y = mx + c



Ch6 (page 54)

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Find the gradient of the line.

Q2. Are the two lines parallel. Explain your answer

y = x + 2 and y = 2x + 2

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1. We are learning to model real life situations using straight line theory

1.1. Be able to work out Be able to work out gradient and y intercept gradient and y intercept using a graph.using a graph.

Modelling Using Modelling Using Straight Line Straight Line

EquationEquation

2.2. Form an equation for any Form an equation for any straight line graph.straight line graph.

Nat 5

A LINEAR equation can and is usually written in the form

Note : Other letters can be used

y = mx + c

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Nat 5

Real – Life Straight Line

The following equations are NOT linear y = mx2 + c

v = ut + a

v = ut2 + aWhy ?

Not allowed powers greater than 1

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Real-Life Straight LineReal-Life Straight Line

310 2

3

1

4

2

6

5

hh 00 11 22 33

PP 00 22 44 66

(0,0)(1,2) (3,6)

(2,4)

P

h

P = 2h

Nat 5

John earns £2 an hour

Using the table draw a graph of P against h

Write down a formula

connecting P and h.

How much did he get paid for

working 48 hours.

Find the gradient and P- intercept

P = 2x48

P = £96

m = 2

c = 0

Modelling Real – life

0 1 2 3 4 5 6 7 8 9 10 T

V

10

20

30

40

50

60

70

80

90

100

Cost

£Time

(hours)

The cost for hiring a plumber per hour is shown below

(a) Calculate the gradient.

(b) What is the value of V when T = 0

(c) Write down an equation connecting V and T.

Pick any 2 points

(7,100)

(0,30)

2 1

2 1

y ym

x x

100 307 0

10

30

V =

30

T +

10

(d) Find the cost for a plumber for 10 hours?

T = 10 C = 10x10 + 30 = £130

80

5-5

Modelling Real – life

0 1 2 3 4 5 6 7 8 9 10 T

V

10

20

30

40

50

60

70

80

90

100

Volu

me

Time (mins)

The graph shows how a the volume of water tank drains over time.

(a) Calculate the gradient.

(b) What is the value of V when T = 0

(c) Write down an equation connecting V and T.

Pick any 2 points

(8,40)

(0,80)

2 1

2 1

y ym

x x

40 808 0

80

V = T +

(d) How long before the tank is empty?

V = 0 0 = -5xT + 80 T = (-80) ÷(–5) = 16mins



Ch6 (page 55)

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Real – Life Real – Life Straight LineStraight Line

Nat 5



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lie on the same line.

Q2. Are the two lines parallel. Explain answer

y = -4x + 1 and 6y - 24x = 12

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1. We are revising gradient. 1.1. Use gradient formula to Use gradient formula to calculate gradient.calculate gradient.

Gradient - RevisionGradient - RevisionNat 5


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2 1

2 1

= m

y ygradient

x x

It is a measure of how steep a line is

A line sloping up from left to right is a positive gradient

A line sloping down from left to right is a negative gradient


Nat 5

Remember parallel means same gradient

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Higher Outcome 1

Apr 20, 2023www.mathsrevision.com34

m < 0

m > 0

m = 0

x = a

y = c

Sloping left to right up has +ve gradient

Sloping left to right down has -ve gradient

Horizontal line has zero gradient.

Vertical line has undefined gradient.

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Nat 5

Gradient - Revision

Find gradients PQ , QR and RP

PQ1

m5

QRm 1RPm 5



Ch6 (page 58)

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Gradient RevisionGradient RevisionNat 5



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lie on the same line.


y = -4x + 1 and 6y - 24x = 12

Nat 5


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1. We are learning alternative way to find straight line equation.

1.1. Know Nat 5 version of Know Nat 5 version of straight line equation straight line equation

y – b = m(x – a)y – b = m(x – a)

Higher Form ofHigher Form ofStraight Line Straight Line

EquationEquation

2.2. Form an straight equation Form an straight equation from information given.from information given.

Nat 5

Apr 20, 2023Apr 20, 2023 www.mathsrevision.comwww.mathsrevision.com


To find the equation of a straight line we need to know

Two points that lie on the line

( x1, y1) and ( x2, y2)

OR

The gradient and a point on the line

m and (a,b)

y = mx + c

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Nat 5

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Nat 5

=

The Equation of the Straight LineThe Equation of the Straight Liney – b = m (x - a)

The equation of any line can be found if we know

the gradient and one point on the line.

O

y

x

x - a

P (x, y)

m

A (a, b)y - by - b

x – a

m =y - b

(x – a)m

Gradient,

mPoint (a,

b)

y – b = m ( x – a ) Point on the line ( a, b )

a x

y

b

Demo

Find the equation of the straight line passing through the points (3, -5) and (6,4).

Solution3

39

365)(4

xxyy

m12

12

y - b = m(x – a)

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Nat 5


Using the point (6,4) and the gradient m = 3 sub into straight line equation


y – b = m(x - a)

y – 4 = 3(x - 6)OR


y – b = m(x - a)

y – (-5) = 3(x - 3)



Ch6 (page 59)

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passing through and (4, 5) and parallel to the line2x + y = 1

Q2. Expand (x - 2)(x2 + 3x + 1)

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1. We are learning the general form for a straight line.

1.1. Know the format for the Know the format for the general form of a straight general form of a straight line.line.

General Equation General Equation of a Straight Lineof a Straight Line

2.2. Be able to rearrange Be able to rearrange general form.general form.

Nat 5

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Straight Line FactsStraight Line Facts

Another version of the straight line general formula is:

ax + by + c = 0

You need to be able to arrange this form to identifyKey information

i.e. Gradient and y–intercept.

Demo

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Simultaneous Equations Nat 5

Straight Lines

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Find the gradient and y-intercept for equations below

x - y – 4 = 0 and 4x + 2y + 12 = 0 Rearrange ( BALANCING METHOD) x – y – 4 = 0

y = x - 4m = 1 c = - 4

Rearrange ( BALANCING METHOD) 4x + 2y + 12 = 0

y = -2x - 3m = -2 c = - 6



Ch6 (page 61)

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Find the gradient of the line.


y = -2x + 1 and y = 2x + 1

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1. How to model real life situations using straight line theory

1.1. Be able to work out Be able to work out gradient and y intercept gradient and y intercept using a graph.using a graph.

Best Fit Best Fit Straight Line Straight Line

EquationEquation

2.2. Form an equation for any Form an equation for any straight line graph.straight line graph.

Nat 5

0 10 20 30 40 50 60 70 80 90 100w

h

20

40

60

80

100

120

140

160

180

200

Heig

ht

(cm

s)Weight (Kgs)

x c = 0

(b) Draw in the best-fit line andfind an equation relating height and weight.

2.572.57

0

Best-Fitting Straight Line

Data was collected on pupils weight and height. Data is plotted below.

(a) Is there correlation between height and weight.Pick any 2 points

(70,180)

2 1

2 1

y ym

x x

180 070 0

h = w +

(d) What height is a pupils who weights 40 kgs?

w = 40 h = 2.57x40 = 102.8 cms

x

(0,0)

Yes, a positive correlation

c = 1818

1.25-1.25

0 1 2 3 4 5 6 7 8 9 10 y

v

2

4

6

8

10

12

14

16

18

20

valu

e £

‘ 0

00

Years

(b) Draw in the best-fit line andfind an equation relating value and year.

Best-Fitting Straight Line

A survey was carried out on the value of cars depending on their age The data is plotted below.

(a) Is there correlation between value and age.Pick any 2 points

(8,8)2 1

2 1

y ym

x x

8 188 0

v = y +

(d) What is the cost of a car after 4years?

y = 4 v = -1.25x 4 + 18 = 13

x

(0,18)Yes, a negative correlation

x

Value = £13 000


Now try Sheet provided

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Nat 5

Best Fit Best Fit Straight Line Straight Line

EquationEquation

straight line

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