The problem involves solving a system of linear equations using substitution. The system provided is:

$x + 3y = -2$ $-3x + y = 6$

To solve this by substitution, follow these steps:

1. Solve one equation for one variable: Start by solving the second equation for $y$: $-3x + y = 6 \quad \Rightarrow \quad y = 6 + 3x$

2. Substitute into the first equation: Now substitute $y = 6 + 3x$ into the first equation: $x + 3(6 + 3x) = -2$ Simplify: $x + 18 + 9x = -2 \quad \Rightarrow \quad 10x + 18 = -2$ Solve for $x$: $10x = -2 - 18 \quad \Rightarrow \quad 10x = -20 \quad \Rightarrow \quad x = -2$

3. Substitute $x = -2$ back into the equation for $y$: $y = 6 + 3(-2) = 6 - 6 = 0$

So, the solution to the system is $(x, y) = (-2, 0)$.

Would you like any further details or explanations?

Here are some related questions to expand on this topic:

1. How do you check if the solution to a system of equations is correct?
2. What other methods, aside from substitution, can be used to solve systems of equations?
3. What happens when the system has no solution or infinitely many solutions?
4. Can substitution be used for systems with more than two equations?
5. How can graphing help visualize the solution to a system of linear equations?

Tip: When solving by substitution, always simplify the equations as much as possible to avoid calculation mistakes.