\begin{align}
  \begin{cases}
    x-y&= 3\\
    -3x+3y &= -9
  \end{cases}
\end{align}

Given the system of equations:

\[
\begin{cases}
x - y = 3 \quad \text{(Equation 1)}\\
-3x + 3y = -9 \quad \text{(Equation 2)}
\end{cases}
\]

### Step 1: Simplify Equation 2
Let's simplify Equation 2 by dividing the entire equation by \(-3\):

\[
x - y = 3 \quad \text{(Simplified Equation 2)}
\]

### Step 2: Compare the two equations
Now we observe that the simplified Equation 2 is identical to Equation 1:

\[
x - y = 3
\]

### Step 3: Interpret the result
Since both equations are identical, this means that the system does not give two independent conditions. Rather, both equations represent the same line in the coordinate plane.

### Conclusion
The system has infinitely many solutions, as any point \((x, y)\) that lies on the line \(x - y = 3\) will satisfy both equations.

The general solution can be written as:
\[
x = y + 3, \quad \text{for any } y \in \mathbb{R}
\]

This means that for any real number \(y\), the corresponding \(x\) value will be \(y + 3\).

Would you like more details or have any questions?

Here are some related questions to explore:
1. How can we tell if a system of linear equations has no solution?
2. What is the geometric interpretation of the solution to a system of linear equations?
3. How do we solve a system of equations using matrix methods?
4. Can we use the substitution method to solve non-linear systems of equations?
5. What are the possible outcomes when solving a system of linear equations (e.g., unique solution, no solution, or infinitely many solutions)?

**Tip:** When simplifying a system of equations, always look for opportunities to identify whether the equations are dependent or represent the same line, as this will determine the nature of the solution set.

Learn how to solve a system of linear equations: x - y = 3 and -3x + 3y = -9 with a detailed step-by-step solution. Understand the concept of dependent equations and the conditions for infinitely many solutions. Suitable for students in Grades 7-9.

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