Question

1. determine if the following system of equations has solutions, infinitely many solutions or exactly one solution.

$5x + y = -3$
$-5x - y = 3$

Explanation:

Step1: Analyze the two equations

We have the system:
$$5x + y = -3$$
$$-5x - y = 3$$

Step2: Add the two equations

Adding the left sides and right sides:
$$(5x + y) + (-5x - y) = -3 + 3$$
$$5x - 5x + y - y = 0$$
$$0 = 0$$

Step3: Interpret the result

The equation $0=0$ is always true, meaning the two original equations are equivalent (one is just the negative of the other). This means every point that satisfies one equation satisfies the other.

Answer:

The system has infinitely many solutions.