This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving 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=− yx and yxx =+− 1072
. Solve the equation.
Solution:
6=− yx
6−= xy
yxx =+− 1072
Substitute into ,
61072
−=+− xxx
01682
=+− xx
0)4)(4( =−− xx
4=x
Substitute 4=x into ,
64 −=y
2−=y
Example 2:
The straight line xy 26 −= intersects the curve 082
=−+ xyy at points A and B. Find the
coordinates of points A and B.
Solution:
xy 26 −=
082
=−+ 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
=−−+− xxx
082636244 22
=−−++− xxxx
028182 2
=+− xx
01492
=+− xx
0)2)(7( =−− xx
7=x or 2=x
Substitute 7=x , 2=x into ,
(i) 7=x
)7(26 −=y (7,-8)
8−=y
(ii) 2=x
)2(26 −=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
=+− kh
5254 22
=+ kh
3)5(2)2( =+− kh
3210 =− hk
2
310 −
=
k
h
Substitute into ,
525)
2
310
(4 22
=+
−
k
k
1 2
1
2
1
2 1
simplify

3. Additional Mathematics Module Form 4
Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis
Page | 43
525)
4
960100
(4 2
2
=+
+−
k
kk
525960100 22
=++− kkk
0460125 2
=+− kk
0)25)(225( =−− kk
0225 =−k or 025 =−k
25
2
=k or
5
2
=k
Substitute
25
2
=k ,
5
2
=k into ,
(i)
25
2
=k
2
3)
25
2
(10 −
=h
10
1
1−=h
(ii)
5
2
=k
2
3)
5
2
(10 −
=h
2
1
=h
Method 2
522
=+ yx
32 =+ yx
yx 23 −=
Substitute into ,
5)23( 22
=+− yy
59124 22
=++− yyy
04125 2
=+− yy
0)2)(25( =−− yy
2
1
2
2 1
simplify

4. Additional Mathematics Module Form 4
Chapter 4- Simultaneous Equations SMK Agama Arau, Perlis
Page | 44
025 =−y or 02 =−y
5
2
=y or 2=y
Substitute
5
2
=y , 2=y into ,
(i)
5
2
=y
)
5
2
(23 −=x
5
1
2=x
(ii) 2=y
)2(23 −=x
1−=x
From the coordinate (-2h, 5k),-2h is the value for x and 5k is the value for y.
Hence,
(i)
5
1
22 =− h 12 −=− 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 −= mxy intersects the curve 22
+= xy at
two different points.
Solution:
72 −= mxy
22
+= xy
2
10
1
1−=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
+=− xmx
0922
=+− mxx
If and intersect at two points,
042
>− acb
0)9)(1(4)2( 2
>−− am
0364 2
>−m
092
>−m
Let
092
=−m
92
=m
3±=m
y
-3 3 m
If 092
>−m , the range of values of m is 3−<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 −=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 =− yx and 1162
+−= xxy .
6. Solve the equation 161232 2
=+=+ xyxyx .
1 2
1 2