### Instruction: If infinitely many values of $y$ satisfy the equation $2(4+cy) = 12y+8$, then what is the value of $c$?### Response: I notice that this equation is linear in $y$, so it either has no solutions, one solution, or infinitely many solutions, depending on the value of $c$.
I want to find the value of $c$ that makes the equation have infinitely many solutions, which means that the equation is true for any value of $y$.
One way to do this is to try to simplify the equation and see if I can eliminate $y$ completely.
I start by expanding the left side and subtracting $12y$ from both sides: $8+2cy-12y = 8$.
Then I factor out $y$ from the terms that contain it: $y(2c-12) = 0$.
Now I see that this equation is true for any value of $y$ as long as $2c-12 = 0$.
Solving for $c$, I get $c = 6$.