Let's solve the system of equations by substitution. The system is:

$-7x - 2y = -38$ $-3y = x$

Step 1: Express $x$ in terms of $y$

From the second equation, we already have $x = -3y$.

Step 2: Substitute $x = -3y$ into the first equation

Substitute $x = -3y$ into the first equation $-7x - 2y = -38$:

$-7(-3y) - 2y = -38$

Simplify the expression:

$21y - 2y = -38$

$19y = -38$

Step 3: Solve for $y$

Divide both sides by 19:

$y = \frac{-38}{19} = -2$

Step 4: Substitute $y = -2$ into $x = -3y$

Now, substitute $y = -2$ into $x = -3y$:

$x = -3(-2) = 6$

Solution:

The solution to the system is $x = 6$ and $y = -2$, or $(6, -2)$.

Would you like further details or explanations on any step?

Here are some related questions to deepen your understanding:

1. How do you graphically interpret the solution of a system of linear equations?
2. What is the method of substitution, and when is it preferable over elimination?
3. How would the solution change if the second equation was $3y = x$ instead of $-3y = x$?
4. What is the significance of having one solution, infinitely many solutions, or no solutions in a system of equations?
5. How can you check if the solution $(6, -2)$ satisfies both equations?

Tip: Always substitute your solution back into the original equations to verify its correctness.