If and are the two solutions to the system of equations above, what is the value of ?

Question

Accepted Answer

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 = x 2 + 2 x + 1. Adding x + 1 to both sides of the equation yields x 2 + 3 x + 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, ( x 1 , y 1 ) = (-2, 1), ( x 2 , y 2 ) = (-1, 0), and y 1 + y 2 = 1 + 0 = 1. Choice A is incorrect. The solutions to the system of equations are ( x 1 , y 1 ) = (-2, 1) and ( x 2 , y 2 ) = (-1, 0). Therefore, -3 is the sum of the x -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.

Question if-and