y = x² + 2x + 1

x + y + 1 = 0

If (x₁, y₁) and (x₂, y₂) are the two solutions to the system of equations above, what is the value of y₁ + y₂?

A. -3

B. -2

C. -1

D. 1

Choice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for y gives y = -x - 1. Substituting the expression -x - 1 for y into the first equation gives -x - 1 = x² + 2x + 1. Adding x + 1 to both sides of the equation yields x² + 3x + 2 = 0. The left-hand side of the equation can be factored by finding two numbers|whose sum is 3 and whose product is 2, which gives (x + 2)(x + 1) = 0. Setting each factor equal to 0 yields x + 2 = 0 and x + 1 = 0, and solving for x yields x = -2 or x = -1. These values of x can be substituted for x in the equation y = -x - 1 to find the corresponding y-values: y = -(-2) - 1 = 2 - 1 = 1 and y = -(-1) - 1 = 1 - 1 = 0. It follows that (-2, 1) and (-1, 0) are the solutions to the given system of equations. Therefore, (x₁, y₁) = (-2, 1),(x₂, y₂) = (-1,0), and y₁ + y₂ = 1 + 0 = 1.

Choice A is incorrect. The solutions to the system of equations are (x₁, y₁) = (-2, 1) and (x₂, y₂) = (-1, 0). Therefore, -3 is the sum of the y-coordinates of the solutions, not the sum of the y-coordinates of the solutions. Choices B and C are incorrect and may be the result of computation or substitution errors.