Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. x, plus, y, equals, minus, 4 x y= −4 minus, x, minus, y, equals, 4 −x−y= 4

Answer 1

The system of equations x + y = -4 and -x - y = 4 has infinitely many solutions, as the equations represent the same line and every point on that line is a solution.

Step-by-step explanation:

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

1. x + y = -4
2. -x - y = 4

We can add equation (1) and equation (2) together: (x + y) + (-x - y) = -4 + 4, which simplifies to 0 = 0. This is a true statement, indicating that the two equations are dependent and represent the same line. Therefore, this system of equations has infinitely many solutions because every point on the line is a solution to the system.

To double-check, we can solve the first equation for y, yielding y = -4 - x, and then substitute this into the second equation. The result is -x - (-4 - x) = 4, which simplifies back to 0 = 0, confirming that the system indeed has infinitely many solutions.