The system of equations provided is:

1. $y = -3x + 4$
2. $x = 2y + 6$

We will solve this system of equations using substitution.

Step 1: Substitute $y = -3x + 4$ into $x = 2y + 6$

Replace $y$ in the second equation: $x = 2(-3x + 4) + 6$

Step 2: Simplify the equation

$x = -6x + 8 + 6$ $x = -6x + 14$

Step 3: Solve for $x$

$x + 6x = 14$ $7x = 14$ $x = 2$

Step 4: Solve for $y$

Substitute $x = 2$ into $y = -3x + 4$: $y = -3(2) + 4$ $y = -6 + 4$ $y = -2$

Final Solution:

$x = 2, \, y = -2$

Let me know if you have questions or want a deeper explanation. Here are related questions:

1. How can we verify the solution for this system of equations?
2. What is the geometric interpretation of these equations?
3. Can this method be used for non-linear systems of equations?
4. What are alternative methods to solve such systems of equations?
5. How does substitution compare to elimination in terms of efficiency?

Tip: Always substitute the solution back into the original equations to ensure accuracy.