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

We are given: $3y - x = -1 \quad (1)$ $x = y + 9 \quad (2)$

Step 1: Substitute equation (2) into equation (1)

Since equation (2) gives us $x = y + 9$, we can substitute $x$ in equation (1):

$3y - (y + 9) = -1$

Step 2: Simplify the equation

Distribute the negative sign and combine like terms:

$3y - y - 9 = -1$ $2y - 9 = -1$

Step 3: Solve for $y$

Add 9 to both sides:

$2y = -1 + 9$ $2y = 8$

Now divide both sides by 2:

$y = \frac{8}{2} = 4$

Step 4: Substitute $y = 4$ back into equation (2)

Now that we know $y = 4$, substitute this value into equation (2) to find $x$:

$x = y + 9 = 4 + 9 = 13$

Final Solution:

The solution to the system of equations is: $x = 13, \quad y = 4$

Would you like more details or have any questions about this solution?

Here are five related questions to expand your understanding:

1. What would happen if you used elimination instead of substitution to solve this system?
2. How can you graphically represent the solution of these two equations?
3. What is the general approach to solving a system of equations using substitution?
4. Can you verify the solution by substituting both values back into the original equations?
5. How does the substitution method work when dealing with more complex nonlinear systems?

Tip: Always substitute back into both original equations to check if your solution satisfies both!