find the value of find the value of a and b for which the following system of equations has infinitely many solutions 2x-(2a+5)y=5 and -(2b+1)x-9y=15
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Comparing 2x+3y=7 with a
1
x+b
1
y+c
1
=0,
we get,
a
1
=2,b
1
=3,c
1
=7
Comparing (a−b)x+(a+b)y=3a+b−2 with a
2
x+b
2
y+c
2
=0,
we get,
a
2
=(a−b),b
2
=(a+b),c
2
=3a+b−2
Given system of equation have infinite solution:
⇒
a
2
a
1
=
b
2
b
1
=
c
2
c
1
⇒
a−b
2
=
a+b
3
=
3a+b−2
7
From
a
2
a
1
=
b
2
b
1
⇒
a−b
2
=
a+b
3
⇒2a+2b=3a−3b
⇒a−5b=0.........(I)
Now from
b
2
b
1
=
c
2
c
1
⇒
a+b
3
=
3a+b−2
7
⇒9a+3b−6=7a+7b
⇒9a−7a+3b−7b=6
⇒2a−4b=6......(II)
Now, for subtracting eq I and eq II
a−5b=0
2a−4b=6
Multiplying eq I by 2
2a−10b=0
−2a+4b=−6
⇒−6b=−6
⇒b=1
Now, substituting b value in eq I
⇒2a−10(1)=0
2a−10=0
2a=10
a=
2
10
a=5
⇒a=5,b=1
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