Question

Determine the number of solutions:
-x + y = 3
-2x + 2y = 3

Answer 1

To determine the number of solutions for the given system of equations:

1. **Write down the equations**:
\[
-x + y = 3 \quad \text{(1)}
\]
\[
-2x + 2y = 3 \quad \text{(2)}
\]

2. **Simplify the second equation (if possible)**:
Divide the second equation by 2:
\[
-x + y = \frac{3}{2} \quad \text{(3)}
\]

3. **Compare equations (1) and (3)**:
\[
-x + y = 3 \quad \text{(1)}
\]
\[
-x + y = \frac{3}{2} \quad \text{(3)}
\]

From equations (1) and (3), we can see that both have the same left-hand side, but different right-hand sides:
\(-x + y = 3\) and \(-x + y = \frac{3}{2}\).

4. **Check for consistency**:
Since \(3\) is not equal to \(\frac{3}{2}\), these two equations are inconsistent.

5. **Conclusion**:
Because the two equations represent parallel lines (having the same slope but different y-intercepts), there are **no solutions** to the system of equations.

Thus, the number of solutions is:
\[
\boxed{0}
\]