7.The sum of three times a number and 6 is 30.Find the
number.
Answers
Step-by-step explanation:
Given let a,b,c be the three numbers
Then according to question
a+b+c=6−−−−(1)
a+3c=7−−−−−(2)
3a+b+c=12−−−−−−−(3)
Using determinant property for finding value of a,b,c we get.
AX=B X=A
−1
B
⎣
⎢
⎢
⎡
1
1
3
1
0
1
1
3
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
a
b
c
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
6
7
12
⎦
⎥
⎥
⎤
where A=
⎣
⎢
⎢
⎡
1
1
3
1
0
1
1
3
1
⎦
⎥
⎥
⎤
Now A
−1
=
∣A∣
adjA
we know
∣A∣=1(0−3)−1(1−9)+1(1−0)
∣A∣=−3+8+1
∣A∣=6
And adj A=
⎣
⎢
⎢
⎡
−3
0
3
+8
−2
−2
1
+2
−1
⎦
⎥
⎥
⎤
adj A=
⎣
⎢
⎢
⎡
−3
8
1
0
−2
2
3
−2
−1
⎦
⎥
⎥
⎤
⇒A
−1
=
6
1
⎣
⎢
⎢
⎡
−3
8
1
0
−2
2
3
−2
−1
⎦
⎥
⎥
⎤
Now
⎣
⎢
⎢
⎡
a
b
c
⎦
⎥
⎥
⎤
=
6
1
⎣
⎢
⎢
⎡
−3
8
1
0
−2
2
3
−2
−1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
6
7
12
⎦
⎥
⎥
⎤
=
6
1
⎣
⎢
⎢
⎡
−18
48
6
+3
−14
+14
6
−24
−12
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
a
b
c
⎦
⎥
⎥
⎤
=
6
1
⎣
⎢
⎢
⎡
18
10
8
⎦
⎥
⎥
⎤
On comparing we get
[a=3,b=
3
5
&c=
3
4
]