Steps to solve this system of equations: x + y = 7 y + x = 11Just wondering easiest step to find values for x and y from the above equations?

Question

Steps to solve this system of equations:      x    +  y  =  7      y    +  x  =  11Just wondering easiest step to find values for   x and   y from the above equations?

Haggar72 · Accepted Answer

Note that:   x  ,  y  ≥  0  (  7  −  y      )    2    =  11  −      y  substitute          y    =  t    , so :  (  7  −      t    2        )    2    =  11  −  t  ⇒      t    4    −  14  ⋅      t    2    +  49  =  11  −  t  ⇒      t    4    −  14  ⋅      t    2    +  t  +  38  =  0Note that possible nonnegative integer solutions of equation           x    +  y  =  7     are :  (  x  ,  y  )  ∈  {  (  1  ,  6  )  ,  (  4  ,  5  )  ,  (  9  ,  4  )  ,  (  16  ,  3  )  ,  (  25  ,  2  )  ,  (  36  ,  1  )  ,  (  49  ,  0  )  }Since pair   (  x  ,  y  )) has to be solution of equation       y    +  x  =  11 also we have that :  (  x  ,  y  )  =  (  9  ,  4  )  ⇒      y    =  2  ⇒  t  =  2   Therefore polynomial          t    4    −  14  ⋅      t    2    +  t  +  38    has to be divisible by   (     t  −  2  )   If you divide this polynomial by   (     t  −  2  )    using Polynomial long division method you will get polynomial :      t    3    +  2      t    2    −  10  t  −  19So you should solve equation          t    3    +  2      t    2    −  10  t  −  19  =  0    to obtain other solutions .This can be done using general formula of roots .

Reed Eaton · Accepted Answer

we can rewrite equations as   (  x  −  4  )  +      y    −  3  =  0 so we have   (      x    −  2  )  (      x    +  2  )  +      y    −  3  =  0 ,  (      y    −  3  )  (      y    +  3  )  +      x    −  2  =  0 substitute       y   from first equation and factor we have   (      x    −  2  )  (  (      x    +  2  )  (      y    +  3  )  −  1  )  =  0 so we have   x  =  4  ,  y  =  9

Steps to solve this system of equations: <msqrt> x </msqrt> + y = 7 <msqrt>

Answered question

Answer & Explanation

New Questions in Algebra I