straight line in real-life equation given any two points gradient revision nat 5 the general...
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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)
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Starter QuestionsStarter Questionsw
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Nat 5
In pairs
“Explain the rules for fractions”
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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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.1
Ch6 (page 50)
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Straight LineStraight LineNat 5
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Starter QuestionsStarter Questionsw
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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.
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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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww
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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
The gradient The gradient using coordinatesusing coordinates
Nat 5
Remember parallel means same gradient
2 1
2 1
y yx
mx
8 2
3 ( 1)m
6
32
m
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y-axis
(3,8) 2 1
2 1
y yx
mx
O x-axis
(1,2)
The gradient The gradient using coordinatesusing coordinates
Find the gradient of the two lines.
8 2
3 1m
6
32
m
(-3,8)
(-1,2)
Nat 5
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The gradient The gradient using coordinatesusing coordinates
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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.2
Ch6 (page 52)
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Straight LineStraight LineNat 5
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Starter QuestionsStarter Questionsw
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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.
Nat 5
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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
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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
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-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
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
Equation of a Straight Line
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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.3
Ch6 (page 54)
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Straight LineStraight LineNat 5
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Starter QuestionsStarter Questionsw
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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
Nat 5
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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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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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.4
Ch6 (page 55)
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Real – Life Real – Life Straight LineStraight Line
Nat 5
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Starter QuestionsStarter Questionsw
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lie on the same line.
Q2. Are the two lines parallel. Explain answer
y = -4x + 1 and 6y - 24x = 12
Nat 5
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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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww
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.com The gradient formula is :
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
The gradient The gradient using coordinatesusing coordinates
Nat 5
Remember parallel means same gradient
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Higher Outcome 1
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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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.5
Ch6 (page 58)
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Gradient RevisionGradient RevisionNat 5
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Starter QuestionsStarter Questionsw
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lie on the same line.
Q2. Are the two lines parallel. Explain answer
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
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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
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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
Equation of a Straight Line
Using the point (6,4) and the gradient m = 3 sub into straight line equation
Equation :y = 3x - 14
y – b = m(x - a)
y – 4 = 3(x - 6)OR
Equation :y = 3x - 14
y – b = m(x - a)
y – (-5) = 3(x - 3)
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.6
Ch6 (page 59)
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Real – Life Real – Life Straight LineStraight Line
Nat 5
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Starter QuestionsStarter Questionsw
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passing through and (4, 5) and parallel to the line2x + y = 1
Q2. Expand (x - 2)(x2 + 3x + 1)
Nat 5
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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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Nat 5
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
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Now try N5 TJ Ex 6.7
Ch6 (page 61)
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Real – Life Real – Life Straight LineStraight Line
Nat 5
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Dept
Starter QuestionsStarter Questionsw
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Find the gradient of the line.
Q2. Are the two lines parallel. Explain answer
y = -2x + 1 and y = 2x + 1
Nat 5
18 Apr 202318 Apr 2023 Created by Mr.Lafferty Maths DeptCreated by Mr.Lafferty Maths Deptww
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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
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Now try Sheet provided
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Nat 5
Best Fit Best Fit Straight Line Straight Line
EquationEquation