To solve the system of equations using the substitution method, we start with the two equations given:

To solve the two equations using substitution, let's start with the equations we have:

1. \( x - 2y = 3 \)
2. \( 4x - 8y = 12 \)

First, we’ll solve the first equation for \( x \):

\[
x = 2y + 3
\]

Now that we have \( x \) all by itself, we can take this \( x \) and plug it into the second equation.

So, let’s put \( (2y + 3) \) into the second equation:

\[
4(2y + 3) - 8y = 12
\]

Next, we multiply \( 4 \) by both \( 2y \) and \( 3 \):

\[
8y + 12 - 8y = 12
\]

Now, you can see that the \( 8y \) and \( -8y \) cancel each other out, which leaves us with:

\[
12 = 12
\]

This is always true! This means that the two equations are really just two different ways of saying the same thing. So, instead of just one solution, we have infinitely many solutions because they represent the same line.

To explain our solution more, we can think of \( y \) as any number. Let's call it \( t \). So we can say:

- \( y = t \) (where \( t \) is any number)

If we put \( t \) back into our equation for \( x \):

\[
x = 2t + 3
\]

So, every pair of \( (x, y) \) that we find can be written like this:

\[
(x, y) = (2t + 3, t)
\]

This means there are an infinite number of answers, all lined up on a straight line.

We can describe that line with the equation:

\[
y = \frac{1}{2}(x - 3)
\]

In conclusion, the system doesn't just have one solution; it has so many solutions that they all make a straight line!