reducing equations to a linear form andrew robertson
TRANSCRIPT
![Page 1: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/1.jpg)
Reducing Equations to a Linear Form
Andrew Robertson
![Page 2: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/2.jpg)
Today
Reducing data relationships to straight lines
WHY ?
One good reason is to be able to make forecasts from data
It is much easier to work and predict future results when the data lies on a straight line
Another reason is that reducing to straight line form may help us understand the relationship between x and y variables better
Consider the following data...............................
![Page 3: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/3.jpg)
![Page 4: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/4.jpg)
Results that give a straight line graphHere’s some data that is linear
x 3.1 5.6 7.0 10.4 12.5
y 2.3 9.7 14.2 24.2 30.4
![Page 5: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/5.jpg)
If these values are plotted on a graph
they lie on a straight line and so obey the law
y = mx + c
![Page 6: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/6.jpg)
Experimental data 1
-15
-10
-5
0
5
10
15
20
25
30
35
-2 0 2 4 6 8 10 12 14
x
![Page 7: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/7.jpg)
• We can calculate a and b from the graph as follows:
The gradient, a Choosing two points a long way apart, the
line passes through (2.3, 0) and (12.0, 28.5),
Therefore gradient a = 28.5/9.7 = 2.94 = 2.9 (to 2 s.f)
The y intercept, b The graph cuts the y axis at (0, -7) , therefore y intercept b = -7
Equation is then y = 2.9x – 7.0
![Page 8: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/8.jpg)
Equations of the form y = ax²+ b
![Page 9: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/9.jpg)
If in another experiment the data appears to satisfy a quadratic relationship; If we let
Y =y, X = x²
Then we can get a straight line by plotting
Y = aX + b
We can plot Y (=y) against X (= x²) we should get a straight line and we can find a and b from our graph.
![Page 10: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/10.jpg)
Example A hosepipe squirts water and the height,
y metres of the water above a fixed level at a distance x m from the hose is measured as
This is thought to obey the law y = ax2 + b
x 2 4 5 6 7 8
y 6.1 3.6 2.2 -0.1 -2.9 -5.5
![Page 11: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/11.jpg)
We need to make a table with values of x² and plot these on a graph, (a BIG graph if by plotted by hand!)
x²=X 4 16 25 36 49 64
y=Y 6.1 3.6 2.2 -0.1 -2.9 -5.5
![Page 12: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/12.jpg)
Hosepipe: height (Y) against distance squared (X)
-8
-6
-4
-2
0
2
4
6
8
0 10 20 30 40 50 60 70
X (distance squared)
Y (h
eigh
t)
![Page 13: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/13.jpg)
Plotting Y against X, gives a straight line Y = aX + b
From the graph, choosing 2 points e.g. (0, 6.9) and (35, 0) gives a = -0.20 (to 2 s.f.) (The gradient is -0.2) The line cuts the Y axis where Y = 6.9
and so b = 6.9
Therefore y = -0.2X + 6.9 or
y = -0.2x2 + 6.9
![Page 14: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/14.jpg)
Hosepipe: height (Y) against distance squared (X)
-8
-6
-4
-2
0
2
4
6
8
0 10 20 30 40 50 60 70
X (distance squared)
Y (h
eigh
t)
![Page 15: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/15.jpg)
Equations of the form y = kxn
(Power Law growth / decay y=kx-n )
![Page 16: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/16.jpg)
y = kxn
• Plot logy against logx
• Intercept is logk
• Gradient is n
![Page 17: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/17.jpg)
A water pipe is being laid between two points. The following data are being used to show how, for a given pressure difference, the rate of flow R litres per second, varies with the pipe diameter d cm
d 1 2 3 5 10
R 0.02 0.32 1.62 12.53 199.8
![Page 18: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/18.jpg)
Here we try the relationship R = kdm where k is a constant
logR = mlogd + logk
Compare with y = mx + c
log d 0 0.3 0.48 0.70 1.00 (x)
log R -1.70 -0.49 0.21 1.10 2.30 (y)
![Page 19: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/19.jpg)
Log flow rate against log pipe diameter
y = 4.0x - 1.7
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2
Log d
Log
R
![Page 20: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/20.jpg)
Log flow rate against log pipe diameter
y = 4.0x - 1.7
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2
Log d
Log
R
![Page 21: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/21.jpg)
Reading from the graph we can see that c = = - 1.70
so k = = 0.02
gradient m = (change in y)/(change in x)
= 4.0
The relationship is then
R = 0.02d4
![Page 22: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/22.jpg)
Equations of the form y = kax
(Exponential relationships)
![Page 23: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/23.jpg)
y = kax
Plot logy against x
intercept is logk
gradient is loga
![Page 24: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/24.jpg)
The temperature θ in ºC of a cup of coffee after t minutes is recorded below
t 2 4 6 8 10 12
θ 81 70 61 52 45 38
![Page 25: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/25.jpg)
• If the relationship is of the form = kat
where k and a are constants
• log = (loga)t + logk
• (y = mx + c)
t 2 4 6 8 10 12
log θ 1.91 1.85 1.79 1.72 1.65 1.58
![Page 26: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/26.jpg)
Log theta against time
y = -0.03x + 1.98
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12 14
t (minutes)
Log
thet
a
![Page 27: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/27.jpg)
Reading from the graph gives logk = 1.98 k = 101.98 so k =
= 95.5
gradient = y / x = -0.03 loga = - 0.03 so a = 10-0.03 = 0.93
Relationship is = 95.5 x 09.3t
![Page 28: Reducing Equations to a Linear Form Andrew Robertson](https://reader033.vdocuments.us/reader033/viewer/2022061508/5697bf731a28abf838c7ee12/html5/thumbnails/28.jpg)
Summary
x
n
kby
kxy
xy
2Plot Y vs X where X=x2
Plot Log(y) vs Log(x)
Plot Log(y) vs x