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The given system of equations is:

\[
\begin{cases}
y = -\frac{1}{3}x - 1 \\
-x - 3y = 0
\end{cases}
\]

### Step 1: Solve the second equation for \(x\) or \(y\).
We can rewrite the second equation:

\[
-x - 3y = 0 \quad \Rightarrow \quad x = -3y
\]

Now we have \(x = -3y\).

### Step 2: Substitute \(x = -3y\) into the first equation.
Using \(x = -3y\) in the first equation:

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

Simplifying:

\[
y = y - 1
\]

This results in:

\[
0 = -1
\]

### Conclusion:
The equation \(0 = -1\) is a contradiction, meaning there is no solution to this system of equations. Therefore, the system has **no solution**.

Would you like further details or clarifications?

Here are some related questions you might explore:
1. How can you determine whether a system of equations is inconsistent?
2. What does it mean geometrically for two lines to have no solution?
3. How do you solve a system of equations graphically?
4. Can systems with parallel lines have solutions?
5. What methods can you use to solve systems of linear equations?

**Tip**: When a system of linear equations has no solution, the lines representing the equations are parallel.

Graph the system below and write its solution:
 y = -1/3x - 1, 
 -x - 3y = 0

Learn how to solve the system of linear equations y = -1/3x - 1 and -x - 3y = 0 graphically. This step-by-step guide explores algebraic methods and highlights the concept of inconsistent systems, which lead to no solution. Suitable for students in Grades 9-11.

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