Question 1106903
<br>
3x+4y=12
x+ky=4<br>
In order for the two equations to have infinitely many solutions, the equations must be equivalent.  Multiply the second equation by 3; now the two equations are
3x+4y=12
3x+3ky=12<br>
For the equations to be equivalent, we must have equal coefficients for y: 3k=4; so k = 4/3.<br>
I will guess that by "listing all solutions" you are looking for parametric equations.  To get them, solve the equation for y and then use x as the parameter:<br>
3x+4y = 12
4y = -3x+12
y = (-3/4)x+3<br>
The parametric equations representing all solutions are
x = t
y = (-3/4)t+3