do the equations y=2x+3 and x=1-2y have common solution, no common solution or infinitely many common solutions?

Answer 1

Y = 2x + 3
Eq1: -2x + y = 3

X = 1 - 2y
Eq2: x + 2y = 1
Multiply Eq2 by 2 and add Eq1 and Eq2:
-2x + y = 3
+2x + 4y = 2
sum: 5y = 5
Y = 1

In Eq2, replace Y with 1 and solve for x:

x + 2*1 = 1
X = -1.

Solution set: (x,y) = (-1,1).
One common solution.