Question

Find the value(s) of p in (i) to (iv) and p and q in (v) for the following pair of equations:

2x + 3y = 7 and 2px + py = 28 – qy,

if the pair of equations have infinitely many solutions.

Answer 1

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

`a_1/a_2 = 2/(2p)`

`b_1/b_2 = 3/(p + q)`

`c_1/c_2 = 1/4`

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

`a_1/a_2 = b_1/b_2 = c_1/c_2`

`1/p = 3/(p + q) = 1/4`

Taking first and third parts, we get

p = 4

Again, taking last two parts, we get

`3/(p + q) = 1/4`

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.