https://file.aiquickdraw.com/mathbot/file/1727187347552vm3lt27v.jpg

The system of equations given in the image is:

\[
\begin{cases}
-x - 3y = 1 \\
-4x - 12y = -1
\end{cases}
\]

To solve this system, we can use either substitution or elimination. Let's simplify and analyze the system to determine the solution.

### Step 1: Simplify the second equation

The second equation is:
\[
-4x - 12y = -1
\]
We can divide the entire equation by \(-4\) to simplify it:
\[
x + 3y = \frac{1}{4}
\]

Now, we have the following system:
\[
\begin{cases}
-x - 3y = 1 \\
x + 3y = \frac{1}{4}
\end{cases}
\]

### Step 2: Add the two equations

Adding both equations together cancels out the variables:

\[
(-x - 3y) + (x + 3y) = 1 + \frac{1}{4}
\]

This simplifies to:
\[
0 = 1 + \frac{1}{4}
\]
\[
0 = \frac{5}{4}
\]

This is a contradiction, meaning the system of equations is inconsistent and has no solutions.

### Conclusion:
The correct answer is **a. no solutions**.

---

Do you have any questions or want further details? Here are 5 related questions to consider:

1. How can we check if a system of equations has no solution using determinants?
2. What are the differences between dependent and inconsistent systems?
3. What is the geometric interpretation of systems of equations with no solutions?
4. How does substitution compare to elimination when solving systems of equations?
5. Can a system of nonlinear equations also have no solution?

**Tip**: When solving systems of equations, always check for contradictions after simplifying to avoid misinterpretation of the results.

Solve the system of equations: -x - 3y = 1 and -4x - 12y = -1

Learn how to solve a system of linear equations using the elimination method. This problem focuses on determining the consistency of the system, identifying if there is no solution, one solution, or infinitely many solutions. Suitable for students in Grades 8-10.

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