Question

find the values of m & n for which the following system of linear equations has infinitely many solutions. 2x+3y=7-------(1) 2mx+(m+n)y=28-------(2)

Answer 1

a1
infinitely many solution = ---- = b1 c1 a2
---- = ----
b2 c2

2 3 7
here, ---- = ---- = ---- ------(1)
2m m+n 28

2 7 1
from(1) eq ---- = ---- = ----
2m 28 4

1 1
---- = ---- , (m= 4) ---(2)
m 4

put eq(2) in equation(1)

3 1
----- = ---- ,
n+4 4

12 = n+4

♡(n = 8)♡
♡(m = 4)♡

hope it help ful for every one.... :)
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Answer 2

This system of equations is of the form

a

1

​

x+b

1

​

y+c

1

​

=0

a

2

​

x+b

2

​

y+c

2

​

=0

where a

1

​

=2m−1,b

1

​

=3,c

1

​

=−5

and a

2

​

=3,b

2

​

=n−1 and c

2

​

=−2

For infinitely many solutions, we must have

a

2

​

a

1

​

​

=

b

2

​

b

1

​

​

=

c

2

​

c

1

​

​

The given system of equations will have an infinite number of solutions, if

3

2m−1

​

=

n−1

3

​

=

−2

−5

​

⇒

3

2m−1

​

=

n−1

3

​

=

2

5

​

⇒

3

2m−1

​

=

2

5

​

 and  

n−1

3

​

=

2

5

​

⇒4m−2=15 and 6=5n−5

⇒4m=17 and 5n=11

⇒m=

4

17

​

 and n=

5

11

​

Hence, the given system of equations will have an infinite number of solutions, if m=

4

17

​

 and n=

5

11

​

.