Algebra Know-It-ALL

5. Solve the following pair of equations as a two-by-two system, including the complex-
   number solutions, if any. Let x be the independent variable, and then define y as a
   function of x in each equation:

x^2 +x−y=− 1

and

x^2 − 2 x−y= 2

6. Check the solution(s) to Prob. 5 in the original equations for correctness.


7. Solve the following pair of equations as a two-by-two system, including the complex-
   number solutions, if any. Let x be the independent variable, and then define y as a
   function of x in each equation:

x^2 +y= 0

and

2 x^3 −y= 0

Explain why one of the solutions has multiplicity 2.

8. Check the solution(s) to Prob. 7 in the original equations for correctness.


9. Solve the following pair of equations as a two-by-two system, including the complex-
   number solutions, if any. Let x be the independent variable, and then define y as a
   function of x in each equation:

4 x^3 + 2 x^2 + 2 x− 2 y− 8 = 0

and

3 x^3 − 2 x^2 + 4 x−y− 5 = 0

What is the multiplicity of each solution?

10. Check the solution(s) to Prob. 9 in the original equations for correctness.

462 More Two-by-Two Systems