Question

find the value for alpha and beta for which the following pair of linear equations has infinite number of solutions : 2x + 3y = 7; 2alpha x + (alpha + beta) y = 28

Answer 1

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.

Answer 2

To find the values of alpha and beta for which the pair of linear equations has an infinite number of solutions, we need to determine when the two equations are dependent or have the same slope. This occurs when the ratio of the coefficients of x and y in both equations is the same. In this case, we can compare the coefficients:

2/2alpha = 3/(alpha + beta)

Simplifying this equation will help us find the values of alpha and beta that satisfy the condition for an infinite number of solutions.