Question

The point (0,-2) is a solution to which system?

-5x + y = -2
-3x + 6y = -12

-5x = y - 3
3x - 8y = 24

-4x + y = 6
-5x - y = 21

x + y = 2
-x + 2y = 16

Answer 1

To determine to which system the point (0, -2) is a solution, we need to substitute \(x = 0\) and \(y = -2\) into each system and check if the equations are satisfied.

Let's examine each system one by one.

1. **System 1:**
\[
-5x + y = -2 \quad \text{(1)}
\]
\[
-3x + 6y = -12 \quad \text{(2)}
\]
Substituting \(x = 0\) and \(y = -2\) in (1):
\[
-5(0) + (-2) = -2 \Rightarrow -2 = -2 \quad \text{(True)}
\]
Substituting in (2):
\[
-3(0) + 6(-2) = -12 \Rightarrow -12 = -12 \quad \text{(True)}
\]
Both equations are satisfied.

2. **System 2:**
\[
-5x = y - 3 \quad \text{(1)}
\]
\[
3x - 8y = 24 \quad \text{(2)}
\]
Substituting \(x = 0\) and \(y = -2\) in (1):
\[
-5(0) = -2 - 3 \Rightarrow 0 = -5 \quad \text{(False)}
\]
(We don't need to check the second equation because the first one is false.)

3. **System 3:**
\[
-4x + y = 6 \quad \text{(1)}
\]
\[
-5x - y = 21 \quad \text{(2)}
\]
Substituting \(x = 0\) and \(y = -2\) in (1):
\[
-4(0) + (-2) = 6 \Rightarrow -2 = 6 \quad \text{(False)}
\]
(We don't need to check the second equation because the first one is false.)

4. **System 4:**
\[
x + y = 2 \quad \text{(1)}
\]
\[
-x + 2y = 16 \quad \text{(2)}
\]
Substituting \(x = 0\) and \(y = -2\) in (1):
\[
0 + (-2) = 2 \Rightarrow -2 = 2 \quad \text{(False)}
\]
(We don't need to check the second equation because the first one is false.)

After evaluating all systems, we find that the point (0, -2) satisfies only the first system.

**Conclusion: The point (0, -2) is a solution to the system:**

\[
-5x + y = -2
\]
\[
-3x + 6y = -12
\]