If the system of equations 2x + 3y = 7 2ax + (a + b)y = 28 hasinfinitely many solutions, then

Given pair of linear equations is

2x + 3y = 7

2px + py = 28 – qy

or 2px + (p + q)y – 28 = 0

On comparing with ax + by + c = 0,

We get,

Here, a1 = 2, b1 = 3, c1 = – 7;

And a2 = 2p, b2 = (p + q), c2 = – 28;

a1/a2 = 2/2p

b1/b2 = 3/ (p+q)

c1/c2 = ¼

Since, the pair of equations has infinitely many solutions i.e., both lines are coincident.

a1/a2 = b1/b2 = c1/c2

1/p = 3/(p+q) = ¼

Taking first and third parts, we get

p = 4

Again, taking last two parts, we get

3/(p+q) = ¼

p + q = 12

Since p = 4

So, q = 8

Here, we see that the values of p = 4 and q = 8 satisfies all three parts.

Hence, the pair of equations has infinitely many solutions for all values of p = 4 and q = 8.

The given system of equations can be written as
2x + 3y - 7 = 0                         ….(i)
2ax + (a + b)y – 28 = 0             ….(ii)
This system is of the form:
`a_1x+b_1y+c_1 = 0`
`a_2x+b_2y+c_2 = 0`
where, `a_1 = 2, b_1= 3, c_1= -7 and a_2 = 2a, b_2 = a + b, c_2= – 28`
For the given system of linear equations to have an infinite number of solutions, we must have:

`(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`
`⇒2/(2a) = 3/(a+b) = (−7)/(−28)`
`⇒ 2/(2a) =( −7)/(−28 )= 1/4 and 3/(a+b) = (−7)/(−28) = 1/4`
⇒ a = 4 and a + b = 12
Substituting a = 4 in a + b = 12, we get
4 + b = 12 ⇒ b = 12 – 4 = 8
Hence, a = 4 and b = 8.

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Updated On: 27-06-2022

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