Solve for integers x, y, z
x + y = 1 - z ,  x^3 + y^3 = 1 - z^2
 
Please also suggest the chapters from where I can prepare for the kinds of questions

Asked by nitishkrnehu09 | 2nd Dec, 2016, 11:44: PM

Expert Answer:

straight x plus straight y equals 1 minus straight z
rightwards double arrow straight x plus straight y plus straight z space equals space 1

straight x cubed plus straight y cubed equals 1 minus straight z squared
rightwards double arrow straight x cubed plus straight y cubed plus straight z squared space equals space 1

therefore straight x plus straight y plus straight z space equals space straight x cubed plus straight y cubed plus straight z squared
straight x cubed plus straight y cubed minus left parenthesis straight x plus straight y right parenthesis plus straight z squared minus straight z equals 0
left parenthesis straight x plus straight y right parenthesis left parenthesis straight x squared plus straight y squared minus xy right parenthesis minus left parenthesis straight x plus straight y right parenthesis plus left parenthesis straight z squared minus straight z right parenthesis equals 0
left parenthesis straight x plus straight y right parenthesis left parenthesis straight x squared plus straight y squared minus xy right parenthesis minus left parenthesis 1 minus straight z right parenthesis plus left parenthesis straight z squared minus straight z right parenthesis space equals space 0

left parenthesis straight x plus straight y right parenthesis left parenthesis straight x squared plus straight y squared minus xy right parenthesis plus left parenthesis straight z squared minus 1 right parenthesis space equals space 0
left parenthesis straight x plus straight y right parenthesis left parenthesis straight x squared plus straight y squared minus xy right parenthesis plus left parenthesis straight z minus 1 right parenthesis left parenthesis straight z plus 1 right parenthesis space equals space 0

rightwards double arrow either space left parenthesis straight x plus straight y right parenthesis left parenthesis straight x squared plus straight y squared minus xy right parenthesis space equals space 0 space space space or space left parenthesis straight z minus 1 right parenthesis left parenthesis straight z plus 1 right parenthesis space equals space 0

if space left parenthesis straight x plus straight y right parenthesis left parenthesis straight x squared plus straight y squared minus xy right parenthesis equals 0
rightwards double arrow straight x equals negative straight y space

if space left parenthesis straight z minus 1 right parenthesis left parenthesis straight z plus 1 right parenthesis space equals space 0
rightwards double arrow straight z space equals space 1 space space space or space space space space straight z space equals space minus 1


The space solution space is space space left parenthesis straight x comma space minus straight x comma space 1 right parenthesis space space or space left parenthesis straight x comma space minus straight x comma space minus 1 right parenthesis. space

Answered by Vijaykumar Wani | 12th Dec, 2016, 12:24: PM