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

Question

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

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

\[
x = 2y + 3
\]

Now that we have \( x \) expressed in terms of \( y \), we can substitute this expression for \( x \) into the second equation:

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

Now, distribute the 4:

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

The \( 8y \) terms cancel each other out:

\[
12 = 12
\]

This statement is always true, which indicates that the two equations are actually equivalent. This means that the system has infinitely many solutions, as they represent the same line.

To express the solution, we can write:

- The variable \( y \) can be any real number \( t \).
- Substituting \( t \) into the expression for \( x \):

\[
x = 2t + 3
\]

Thus, the solutions can be expressed as:

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

where \( t \) is any real number. Therefore, the solution set is the line defined by:

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

In summary, there are infinitely many solutions to the system, represented by the equations of the line
make thislike a 6th grader wrote it

Answer 1

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!