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8. Solution of Simultaneous Linear Equations
Exercise 8.1
1 A. Question
Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 30 – 28 = 2
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 6 = 6
C
12
 = (– 1)
1 + 2
 4 = – 4
C
21
 = (– 1)
2 + 1
 7 = – 7
C
22
 = (– 1)
2 + 2
 5 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 
1 B. Question
Solve the following system of equations by matrix method:
Page 2


8. Solution of Simultaneous Linear Equations
Exercise 8.1
1 A. Question
Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 30 – 28 = 2
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 6 = 6
C
12
 = (– 1)
1 + 2
 4 = – 4
C
21
 = (– 1)
2 + 1
 7 = – 7
C
22
 = (– 1)
2 + 2
 5 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 
1 B. Question
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 10 – 6 = 4
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 2 = 2
C
12
 = (– 1)
1 + 2
 3 = – 3
C
21
 = (– 1)
2 + 1
 2 = – 2
C
22
 = (– 1)
2 + 2
 2 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 4
1 C. Question
Solve the following system of equations by matrix method:
3x + 4y = 5
x – y = – 3
Answer
The above system of equations can be written as
Page 3


8. Solution of Simultaneous Linear Equations
Exercise 8.1
1 A. Question
Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 30 – 28 = 2
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 6 = 6
C
12
 = (– 1)
1 + 2
 4 = – 4
C
21
 = (– 1)
2 + 1
 7 = – 7
C
22
 = (– 1)
2 + 2
 5 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 
1 B. Question
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 10 – 6 = 4
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 2 = 2
C
12
 = (– 1)
1 + 2
 3 = – 3
C
21
 = (– 1)
2 + 1
 2 = – 2
C
22
 = (– 1)
2 + 2
 2 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 4
1 C. Question
Solve the following system of equations by matrix method:
3x + 4y = 5
x – y = – 3
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = – 3 – 4 = – 7
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 – 1 = – 1
C
12
 = (– 1)
1 + 2
 1 = – 1
C
21
 = (– 1)
2 + 1
 4 = – 4
C
22
 = (– 1)
2 + 2
 3 = 3
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X = 1 Y = – 2
1 D. Question
Solve the following system of equations by matrix method:
3x + y = 19
3x – y = 23
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = – 3 – 3 = – 6
Page 4


8. Solution of Simultaneous Linear Equations
Exercise 8.1
1 A. Question
Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 30 – 28 = 2
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 6 = 6
C
12
 = (– 1)
1 + 2
 4 = – 4
C
21
 = (– 1)
2 + 1
 7 = – 7
C
22
 = (– 1)
2 + 2
 5 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 
1 B. Question
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 10 – 6 = 4
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 2 = 2
C
12
 = (– 1)
1 + 2
 3 = – 3
C
21
 = (– 1)
2 + 1
 2 = – 2
C
22
 = (– 1)
2 + 2
 2 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 4
1 C. Question
Solve the following system of equations by matrix method:
3x + 4y = 5
x – y = – 3
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = – 3 – 4 = – 7
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 – 1 = – 1
C
12
 = (– 1)
1 + 2
 1 = – 1
C
21
 = (– 1)
2 + 1
 4 = – 4
C
22
 = (– 1)
2 + 2
 3 = 3
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X = 1 Y = – 2
1 D. Question
Solve the following system of equations by matrix method:
3x + y = 19
3x – y = 23
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = – 3 – 3 = – 6
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 – 1 = – 1
C
12
 = (– 1)
1 + 2
 3 = – 3
C
21
 = (– 1)
2 + 1
 1 = – 1
C
22
 = (– 1)
2 + 2
 3 = 3
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X = 7 Y = – 2
1 E. Question
Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = – 1
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 6 – 7 = – 1
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 2 = 2
Page 5


8. Solution of Simultaneous Linear Equations
Exercise 8.1
1 A. Question
Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 30 – 28 = 2
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 6 = 6
C
12
 = (– 1)
1 + 2
 4 = – 4
C
21
 = (– 1)
2 + 1
 7 = – 7
C
22
 = (– 1)
2 + 2
 5 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 
1 B. Question
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 10 – 6 = 4
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 2 = 2
C
12
 = (– 1)
1 + 2
 3 = – 3
C
21
 = (– 1)
2 + 1
 2 = – 2
C
22
 = (– 1)
2 + 2
 2 = 5
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X =  Y = 4
1 C. Question
Solve the following system of equations by matrix method:
3x + 4y = 5
x – y = – 3
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = – 3 – 4 = – 7
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 – 1 = – 1
C
12
 = (– 1)
1 + 2
 1 = – 1
C
21
 = (– 1)
2 + 1
 4 = – 4
C
22
 = (– 1)
2 + 2
 3 = 3
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X = 1 Y = – 2
1 D. Question
Solve the following system of equations by matrix method:
3x + y = 19
3x – y = 23
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = – 3 – 3 = – 6
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 – 1 = – 1
C
12
 = (– 1)
1 + 2
 3 = – 3
C
21
 = (– 1)
2 + 1
 1 = – 1
C
22
 = (– 1)
2 + 2
 3 = 3
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X = 7 Y = – 2
1 E. Question
Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = – 1
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 6 – 7 = – 1
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 2 = 2
C
12
 = (– 1)
1 + 2
 1 = – 1
C
21
 = (– 1)
2 + 1
 7 = – 7
C
22
 = (– 1)
2 + 2
 3 = 3
Also, adj A = 
= 
A 
– 1
 = 
A 
– 1
 = 
Now, X = A 
– 1
B
Hence, X = – 15 Y = 7
1 F. Question
Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12
Answer
The above system of equations can be written as
 or AX = B
Where A =  B =  and X = 
|A| = 9 – 5 = 4
So, the above system has a unique solution, given by
X = A 
– 1
B
Let C
ij
 be the cofactor of a
ij
in A, then
C
11
 = (– 1)
1 + 1
 3 = 3
C
12
 = (– 1)
1 + 2
 5 = – 5
C
21
 = (– 1)
2 + 1
 1 = – 1
C
22
 = (– 1)
2 + 2
 3 = 3

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