Solving a system of equations, why aren't the solutions preserved? 6 x 2 + 8 x y + 4 y 2 = 3and 2 x 2 + 5 x y + 3 y 2 = 2Multiply the second by 8 to get: 16 x 2 + 40 x y + 24 y 2 = 16Multiply the first by 5 to get: 30 x 2 + 40 x y + 20 y 2 = 15Subtract the two to get: 14 x 2 − 4 y 2 = − 1Later, the guy said to disregard his solution because the solutions to the first two equations do not satisfy the third equation. Why does this happen?

Question

Solving a system of equations, why aren&#039;t the solutions preserved?  6      x    2    +  8  x  y  +  4      y    2    =  3and  2      x    2    +  5  x  y  +  3      y    2    =  2Multiply the second by   8 to get:   16      x    2    +  40  x  y  +  24      y    2    =  16Multiply the first by   5 to get:   30      x    2    +  40  x  y  +  20      y    2    =  15Subtract the two to get:   14      x    2    −  4      y    2    =  −  1Later, the guy said to disregard his solution because the solutions to the first two equations do not satisfy the third equation. Why does this happen?

Jamarcus Shields · Accepted Answer

It doesn&#039;t happen! And in fact, you&#039;ve misquoted the answer by user &#039;response&#039; in that post. It says:my suggestion does not work because a solution to the third eqn need not satisfy the original two eqns.Necessarily, any solution to the first two equations also satisfies the third.What they actually meant is this: there can certainly be solutions to the third equation which do not satisfy the first two. This is because there has been some loss of information - you&#039;ve canceled out terms!For instance, in this particular case,   (  x  ,  y  )  =  (  0  ,      1    2    ) solves the third equation, but not the first two.

Ellen Chang · Accepted Answer

Your initial two equations are (rotated) ellipses; their intersection can be up to four points. Your derived equation is a hyperbola, which has infinitely many solutions, including not only the four but many others.

Solving a system of equations, why aren't the solutions preserved? 6 x 2 </msup>

Answered question

Answer & Explanation

New Questions in Algebra I