Solve the following system of equations { y 2 − 8 x + 9 − x y − 6 x + 12 3 = 1 2 ( x − y ) 2 + 10 x − 6 y + 12 − y = x + 2

Question

Solve the following system of equations            {                                                                  y                2                            −              8              x              +              9                        −                                          x                y                −                6                x                +                12                            3                        =            1                                                              2              (              x              −              y                              )                2                            +              10              x              −              6              y              +              12                        −                          y                        =                          x              +              2

Jarrett Reyes · Accepted Answer

Let       x    ′    =  x  +  1 and       y    ′    =  y  −  1, the system of equations becomes      {                                                      y                              ′                2                                      −            8                          x              ′                        +            2                          y              ′                        +            18                    −                                                    x                ′                                            y                ′                            −              5                              x                ′                            −                              y                ′                            +              17                        3                    =          1                                                  2            (                          x              ′                        −                          y              ′                                      )              2                        +            2                          x              ′                        +            2                          y              ′                        +            4                    −                                    y              ′                        +            1                    =                                    x              ′                        +            1                                  Solving the second equation, we have that      2    (          x      ′        −          y      ′              )      2        +    2          x      ′        +    2          y      ′        +    4    ≥      2    [    (          x      ′        +    1    )    +    (          y      ′        +    1    )    ]    ≥            x      ′        +    1    +            y      ′        +    1  The equality sign occurs when       x    ′    =      y    ′  . Substituting       x    ′    =      y    ′   into the first equation, we have that            x              ′        2              −    6          x      ′        +    18    −                    x                  ′          2                    −      6              x        ′            +      17        3    =  1Let                     x                  ′          2                    −      6              x        ′            +      17        3    =  t  (  t  ≥  2  ), we have that            t      3        +    1    −  t  =  1    ⟺              t      2        −    t    +    1    =      t    +    1      ⟺        t    2    −  2  t  =  0    ⟹    t  =  2      x          ′      2        −  6      x    ′    +  9  =  0    ⟹        x    ′    =      y    ′    =  3    ⟹    (  x  ,  y  )  =  (  2  ,  4  )

reryfaikear · Accepted Answer

Hint: Squaring two times and factorizing your second equation we get      (    4                  x          2        −    8        x    y    +    4                  y          2        +    20        x    −    12        y    +    25    )              (      2      +      x      −      y      )          2    =  0This can be solved for   x or   y and you can eliminate one variable.

Solve the following system of equations <mstyle mathsize="1.2em"> { <mtable

Answered question

Answer & Explanation

New Questions in Algebra I