Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
2x+y=−3
−2x−y=5

Answer 1

To determine if the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we can analyze the equations:

1. \( 2x + y = -3 \) (Equation 1)
2. \( -2x - y = 5 \) (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:

1. \( 2x + y = -3 \)
2. \( 2x + y = -5 \)

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.