The system of equations provided is:

$1. \ 4x + 3y = 8$ $2. \ 4x + 6y = 20$

Step 1: Eliminate one variable

We can eliminate $x$ by subtracting the first equation from the second.

$(4x + 6y) - (4x + 3y) = 20 - 8$

This simplifies to:

$6y - 3y = 12 \quad \Rightarrow \quad 3y = 12$

Solve for $y$:

$y = \frac{12}{3} = 4$

Step 2: Substitute $y = 4$ into one of the original equations

Substitute $y = 4$ into the first equation $4x + 3y = 8$:

$4x + 3(4) = 8$

This simplifies to:

$4x + 12 = 8 \quad \Rightarrow \quad 4x = 8 - 12 \quad \Rightarrow \quad 4x = -4$

Solve for $x$:

$x = \frac{-4}{4} = -1$

Final Solution:

The solution to the system of equations is: $x = -1, \quad y = 4$

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. What happens if we solve this system graphically? What do the lines represent?
2. How do you solve a system of equations using substitution instead of elimination?
3. What would happen if both equations represented the same line?
4. How can we solve a system of three variables using elimination?
5. What are the possible outcomes when solving a system of linear equations (one solution, no solution, infinitely many solutions)?

Tip: Always check your solution by plugging the values of $x$ and $y$ back into both original equations to ensure they satisfy both!