To solve the system of equations using substitution, let's proceed step by step.

The system of equations is:

1. $12x - 3y = -3$
2. $6x + y = 1$

Step 1: Solve one equation for one variable

From the second equation ($6x + y = 1$), solve for $y$:
$y = 1 - 6x$

Step 2: Substitute $y = 1 - 6x$ into the first equation

Substitute $y = 1 - 6x$ into $12x - 3y = -3$:
$12x - 3(1 - 6x) = -3$

Simplify:
$12x - 3 + 18x = -3$
$30x - 3 = -3$
$30x = 0$
$x = 0$

Step 3: Solve for $y$

Now substitute $x = 0$ into $y = 1 - 6x$:
$y = 1 - 6(0)$
$y = 1$

Final Solution:

The solution to the system of equations is:
$x = 0, \, y = 1$

Would you like me to verify this solution or explain any specific steps?

Here are 5 related questions to explore further:

1. How do you solve systems of equations using the elimination method?
2. Can systems of equations have no solution or infinitely many solutions?
3. How would you graphically represent the solution to this system?
4. How do you check the solution of a system of equations?
5. How is substitution different from other methods like elimination?

Tip: Always simplify one equation first to make substitution easier.