To determine which system of equations the point \((0, -2)\) is a solution to, we will substitute \(x = 0\) and \(y = -2\) into each set of equations and verify if both equations in that system are satisfied.
-
System 1:
- \(-x + y = 2\) \[ -0 + (-2) = -2 \quad \text{(not satisfied)} \]
- \(-x + 2y = 16\) \[ -0 + 2(-2) = -4 \quad \text{(not satisfied)} \]
This system does not contain the point.
-
System 2:
- \(-5x - y = 21\) \[ -5(0) - (-2) = 2 \quad \text{(not satisfied)} \]
- \(-4x + y = 6\) \[ -4(0) + (-2) = -2 \quad \text{(not satisfied)} \]
This system does not contain the point.
-
System 3:
- \(-5x = y - 3\) \[ -5(0) = -2 - 3 \implies 0 = -5 \quad \text{(not satisfied)} \]
- \(3x - 8y = 24\) \[ 3(0) - 8(-2) = 16 \quad \text{(not satisfied)} \]
This system does not contain the point.
-
System 4:
- \(-5x + y = -2\) \[ -5(0) + (-2) = -2 \quad \text{(satisfied)} \]
- \(-3x + 6y = -12\) \[ -3(0) + 6(-2) = -12 \quad \text{(satisfied)} \]
This system does contain the point.
Thus, the correct system of equations that includes the point \((0, -2)\) is:
\[ -5x + y = -2 \] \[ -3x + 6y = -12 \]
So, the answer is -5x + y = -2 and -3x + 6y = -12.