For which values of p and q, will the following pair of linear equations have infinitely many

solutions?

4x + 5y = 2; (2p + 7q) x + (p + 8q) y = 2q - p + 1.

In order to have infinitely many solutions with linear equations/functions, the two equations have to be the same;

In accordance, we can say:

(2p + 7q) x = 4x [1]

(p + 8q) y = 5y [2]

2q - p + 1 = 2 [3]

All we have to do is choose two equations and solve them simultaneously (The simplest ones for what I'm doing and hence the ones I'm going to use are [3] and [2]):

Rearrange in terms of p:

p + 8q = 5 [2]

p = 5 - 8q [2]

p + 2 = 2q + 1 [3]

p = 2q - 1 [3]

Now equate rearranged [2] and [3] and solve for q:

5 - 8q = 2q - 1

10q = 6

q = 6/10 = 3/5 = 0.6

Now, substitute q-value into rearranges equations [2] or [3] to get p:

p = 2 (3/5) - 1

p = 6/5 - 1

p = 1/5 = 0.2