Find p and q for which the linear eqn has infinite solutions

6x - (2p-3) y-2q-3=0, 12x - (2p-1) y-5q+1=0

p = 2.5

q = 7

Step-by-step explanation:

The lines should overlap to have infinite solutions, slopes should be same and y-intercepts should be same.

Equations in slope - intercept form:

6x - (2p-3) y-2q-3=0 ⇒ (2p-3) y = 6x - 2q-3 ⇒ y = 6 / (2p-3) x - (2q+3) / (2p-3)

12x - (2p-1) y-5q+1=0 ⇒ (2p-1) y = 12x - 5q+1 ⇒ y=12 / (2p-1) x - (5q-1) / (2p-1)

Slopes equal:

6 / (2p-3) = 12 / (2p-1)

6 (2p-1) = 12 (2p-3)

12p - 6 = 24p - 36

12p = 30

p = 30/12

p = 2.5

y-intercepts equal:

(2q+3) / (2p-3) = (5q-1) / (2p-1)

(2q+3) / (2*2.5-3) = (5q-1) / (2*2.5-1)

(2q+3) / 2 = (5q-1) / 4

4 (2q+3) = 2 (5q-1)

8q+12 = 10q - 2

2q = 14

q = 7