Question

. Find the values of a and b for which the following system of the linear equation has infinitely many solutions (a +b)x -2by = 5a + 2b + I and 3x – y = 14

Answer 1

ANSWER

he given system of equations is

2x+3y−7=0

2αx+(α+β)y−28=0

This system of equation 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

=2,b

1

=3,c

1

=−7

and a

2

=2α,b

2

=(α+β) and c

2

=−28

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 infinite number of solutions, if

2α

2

=

α+β

3

=

−28

−7

⇒

α

1

=

α+β

3

=

4

1

⇒

α

1

=

4

1

and

α+β

3

=

4

1

⇒α=4 and α+β=12

⇒α=4 and β=8

Hence, the given system of equations will have infinitely many solutions, if α=4 and β=8.