To solve the system of equations:

$x = 4y + 8$ $x = 2y + 6$

Step 1: Set the two equations equal to each other

Since both equations are equal to $x$, we can set the right-hand sides of the equations equal to each other:

$4y + 8 = 2y + 6$

Step 2: Solve for $y$

Simplify and solve for $y$:

$4y - 2y = 6 - 8$ $2y = -2$ $y = -1$

Step 3: Solve for $x$

Substitute $y = -1$ into either equation to find $x$. Using the first equation:

$x = 4y + 8$ $x = 4(-1) + 8$ $x = -4 + 8$ $x = 4$

Final Solution:

$x = 4, \, y = -1$

Let me know if you'd like further clarification or a detailed breakdown! Here are some related questions you might explore:

1. How can this solution be represented graphically?
2. How would the solution change if the second equation were $x = 2y + k$?
3. What if the two equations represented parallel lines? How would that affect the solution?
4. What happens if we add a third equation to the system?
5. How does solving a system of equations relate to matrices?

Tip: Always verify your solution by substituting back into both equations!