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The Cavite Mutiny of 1872 was an uprising of military personnel of Fort San Felipe, the Spanish arsenal in Cavite, Philippines on January 20, 1872, about 200 Filipino military personnel of Fort San Felipe Arsenal in Cavite, Philippines, staged a mutiny which in a way led to the Philippine Revolution in 1896. The 1872 Cavite Mutiny was precipitated by the removal of long-standing personal benefits to the workers such as tax (tribute) and forced labor exemptions on order from the Governor General Rafael de Izquierdo. Many scholars believe that the Cavite Mutiny of 1872 was the beginning of Filipino nationalism that would eventually lead to the Philippine Revolution of 1896.
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Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
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In a second workflow supporting the same use case, youβll see:
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And...
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The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
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Learn about:
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β’ Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
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Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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Monitoring Java Application Security with JDK Tools and JFR Events
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Chapter 4 simultaneous equations
1. Additional Mathematics Module Form 4
Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis
Page | 41
CHAPTER 4- SIMULATANEOUS EQUATIONS
4.1 INTRODUCTION
1. In Form Three, we have learned about to solve simultaneous equations between two linear equations
with two unknowns.
2. In Form Four, we will learn about to solve simultaneous equations between linear equation and non-
linear equation which is quadratic equation with two unknowns.
4.2 SOLVING SIMULTANOUS EQUATIONS
Example 1 :
Given 6=β yx and yxx =+β 1072
. Solve the equation.
Solution:
6=β yx
6β= xy
yxx =+β 1072
Substitute into ,
61072
β=+β xxx
01682
=+β xx
0)4)(4( =ββ xx
4=x
Substitute 4=x into ,
64 β=y
2β=y
Example 2:
The straight line xy 26 β= intersects the curve 082
=β+ xyy at points A and B. Find the
coordinates of points A and B.
Solution:
xy 26 β=
082
=β+ xyy
1
2
1 2
1
1
2
2. Additional Mathematics Module Form 4
Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis
Page | 42
Substitute into ,
08)26()26( 2
=ββ+β xxx
082636244 22
=ββ++β xxxx
028182 2
=+β xx
01492
=+β xx
0)2)(7( =ββ xx
7=x or 2=x
Substitute 7=x , 2=x into ,
(i) 7=x
)7(26 β=y (7,-8)
8β=y
(ii) 2=x
)2(26 β=y (2, 2)
2=y
The coordinates of point A and point B are (2, 2) and (7,-8)
Example 3:
Given A (-2h, 5k) is the solution to the simultaneous equations 522
=+ yx and 32 =+ yx , find the
values of h and k.
Solution:
Method 1
From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y. Hence change the terms x
and y with -2h and 5k respectively.
5)5()2( 22
=+β kh
5254 22
=+ kh
3)5(2)2( =+β kh
3210 =β hk
2
310 β
=
k
h
Substitute into ,
525)
2
310
(4 22
=+
β
k
k
1 2
1
2
1
2 1
simplify
4. Additional Mathematics Module Form 4
Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis
Page | 44
025 =βy or 02 =βy
5
2
=y or 2=y
Substitute
5
2
=y , 2=y into ,
(i)
5
2
=y
)
5
2
(23 β=x
5
1
2=x
(ii) 2=y
)2(23 β=x
1β=x
From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y.
Hence,
(i)
5
1
22 =β h 12 β=β h
2
1
=h
(ii)
5
2
5 =k 25 =k
25
2
=k
5
2
=k
Example 4:
Find the range of values of m so that the straight line 72 β= mxy intersects the curve 22
+= xy at
two different points.
Solution:
72 β= mxy
22
+= xy
2
10
1
1β=h
Compare the actual
value of x with the given
1
2
5. Additional Mathematics Module Form 4
Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis
Page | 45
Substitute into
272 2
+=β xmx
0922
=+β mxx
If and intersect at two points,
042
>β acb
0)9)(1(4)2( 2
>ββ am
0364 2
>βm
092
>βm
Let
092
=βm
92
=m
3Β±=m
y
-3 3 m
If 092
>βm , the range of values of m is 3β<m , 3>m .
CHAPTER REVIEW EXERCISE
1. Solve the following simultaneous equations:
(a) 12=+ yx and 982
+= xy (b) 113 =+ qp and 2)2)(1( =++ qp
2. Find the coordinates of the points of intersection of the straight line xy β=102 and the curve
yxy 72 2
=+ .
3. The sum of two numbers is 7 and the sum of the square of two numbers is 29. Find the two numbers.
4. Given that (2, 1) is one of the solutions of the simultaneous equations 42 =+ pyx and
1122
=++ xkyx , where p and k are constants, find the value of p and of k and the other solution.
5. Solve the simultaneous equations 52 =β yx and 1162
+β= xxy .
6. Solve the equation 161232 2
=+=+ xyxyx .
1 2
1 2