To determine if the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we can analyze the equations:
- \( 2x + y = -3 \) (Equation 1)
- \( -2x - y = 5 \) (Equation 2)
Let's first rewrite Equation 2 in a more standard form:
\(-2x - y = 5\) can be rewritten as \( 2x + y = -5\) (by multiplying the entire equation by -1).
Now we have the two equations:
- \( 2x + y = -3 \)
- \( 2x + y = -5 \)
Next, we observe the two equations:
The left-hand sides of both equations are the same (\(2x + y\)) but the right-hand sides are different (\(-3\) and \(-5\)). This indicates that two parallel lines are represented by these equations, which means they will never intersect.
Since the lines do not intersect, the system of equations has no solutions.
Answer: No solutions.