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To form the augmented matrix from the given system of equations, we first need to rewrite the system in standard form, where the terms involving variables \(x\) and \(y\) are on the left-hand side and the constants are on the right-hand side.

The given system of equations is:

\[
1. \quad x - 3y = 0
\]
\[
2. \quad -1 + 3y = -4x
\]

### Step 1: Rearrange the second equation

We will rearrange the second equation to bring all terms involving variables to the left-hand side:

\[
-4x + 3y = 1
\]

### Step 2: Write the system in standard form

Now, the system of equations becomes:
\[
1. \quad x - 3y = 0
\]
\[
2. \quad -4x + 3y = 1
\]

### Step 3: Form the augmented matrix

The coefficients of \(x\) and \(y\) in the equations, along with the constants, will form the augmented matrix:

\[
\begin{pmatrix}
1 & -3 & 0 \\
-4 & 3 & 1
\end{pmatrix}
\]

This is the augmented matrix for the system of equations.

Would you like further clarification or details on solving systems using augmented matrices?

Here are 5 follow-up questions to deepen your understanding:
1. What is the next step after forming an augmented matrix to solve the system of equations?
2. How does Gaussian elimination work on augmented matrices?
3. What are the possible outcomes when solving a system of equations using augmented matrices?
4. Can augmented matrices be used for systems with more than two variables?
5. What is the difference between row echelon form and reduced row echelon form?

**Tip**: Always ensure the system is in standard form before forming the augmented matrix.

Write the augmented matrix for the following system of equations:

x - 3y = 0

-1 + 3y = -4x

Learn how to form the augmented matrix for a system of linear equations, such as x - 3y = 0 and -1 + 3y = -4x. This solution covers key concepts in linear algebra, including augmented matrices and Gaussian elimination, suitable for students in Grades 9-12.

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