Question

For what value of n and n the following system of linear equation has infinitely many solutions 3x +4y=12
(m+n)x+2 (m-n)y=5m-1

Answer 1

For infinitely many solutions,

$\frac{a1} {a2}$ = $\frac{b1} {b2}$= $\frac{c1} {c2}$

Here, a1 = 3, a2 = ( m + n ) , b1 = 4 , b2 = 2( m - n ), C1 = 12 , C2 = 5m - 1

$\frac{3} {m+n}$ = $\frac{4} {2(m-n) }$ = $\frac{12} {5m - 1}$

Cross multiplying $\frac{a1} {a2}$ to $\frac{c1} {c2}$

3 ( 5m-1 ) = ( m + n) (12)

15m - 3 = 12m + 12n

15m - 12m - 12n - 3 = 0

3m - 12n - 3 = 0

3( m - 4n - 1 ) = 0

m - 4n - 1 = 0 ---> ( i )

Cross multiplying $\frac{a1} {a2}$ to $\frac{b1} {b2}$

3 ( 2m - 2n) = 4 ( m+n )

6m - 6n = 4m + 4n

6m - 4m - 6n - 4n = 0

2m - 10n = 0

2( m - 5n ) = 0

m - 5n = 0 ---> ( ii )

m = 5n

Putting value of m in equation ( i ),

m - 4n - 1 = 0

5n - 4n = 1

n = 1

Putting value of n in m,

m = 5n

m = 5 × 1 = 5

$<h4> m = 5 and n = 1$