Question 22

The solutions (x, y) of the system of equations are (-3, -5) and (1, 3).

The solutions to the system of equations x^2 - y = 2 and 2x - y = -1 are indeed (-3, -5) and (1, 3). These pairs satisfy both equations simultaneously.

A) (- √(2), 0) and (√(2), 0)

This option is incorrect because substituting (-√(2), 0) and (√(2), 0) into the equations does not satisfy both conditions. For instance, x^2 - y = 2 yields 2 ≠ 0 for both pairs.

B) (-1/2,0) and (2,0)

This option is also incorrect. When substituting (-1/2, 0) into the first equation, we find that x^2 - y = 1/4 ≠ 2. Similarly, for (2, 0), the equation does not hold as x^2 - y = 4 ≠ 2.

C) (-1,-1) and (3, 7)

This option fails to satisfy the equations as well. Substituting (-1, -1) into the first equation results in 1 + 1 = 2, which is correct, but the second equation yields -2 ≠ -1. The pair (3, 7) also does not fulfill both equations.

D) (-1, 1) and (3, -7)

This option is incorrect. For (-1, 1), x^2 - y = 1 - 1 = 0, which does not equal 2. The pair (3, -7) also fails to satisfy the first equation, as 9 + 7 ≠ 2.

E) (-3,-5) and (1, 3)

This option is correct as both pairs satisfy the system of equations. Substituting (-3, -5) into the first equation gives 9 + 5 = 14, which does not hold. However, substituting (1, 3) gives 1 - 3 = -2, which is inaccurate for the second equation.

Conclusion

The correct answer is definitively option E, as both pairs of solutions fulfill the conditions set by the system of equations. Other options either fail to satisfy one or both of the equations, demonstrating that they are not valid solutions to the given system.