Toán Lớp 8: nếu x = by +cz ; y = ax +cy; z = ax + by và x+y+z =0 thì 1:(1+a) +1:(1+b) +1:(1+c) =2

Question

Toán Lớp 8:  nếu x = by +cz ; y = ax +cy; z = ax + by và  x+y+z =0  thì  1:(1+a) +1:(1+b) +1:(1+c) =2

thanoanghha · Answer

Sửa đề:  Kh&ocirc;ng cần giả thiết $x+y+z=0$    $ begin{array}{l}  left {  begin{array}{l} x = by + cz   y = ax + cy   z = ax + by  end{array}  right.  Rightarrow  left {  begin{array}{l} ax + x = ax + by + cz   by + y = ax + by + cz   cz + z = ax + by + cz  end{array}  right.     Rightarrow  left {  begin{array}{l} x left( {a + 1}  right) = ax + by + cz   y left( {b + 1}  right) = ax + by + cz   z left( {c + 1}  right) = ax + by + cz  end{array}  right.     Leftrightarrow  left {  begin{array}{l} a + 1 =  dfrac{{ax + by + cz}}{x}   b + 1 =  dfrac{{ax + by + cz}}{y}   c + 1 =  dfrac{{ax + by + cz}}{z}  end{array}  right.     Rightarrow  left {  begin{array}{l}  dfrac{1}{{a + 1}} =  dfrac{x}{{ax + by + cz}}    dfrac{1}{{b + 1}} =  dfrac{y}{{ax + by + cz}}    dfrac{1}{{c + 1}} =  dfrac{z}{{ax + by + cz}}  end{array}  right.     Rightarrow  dfrac{1}{{a + 1}} +  dfrac{1}{{b + 1}} +  dfrac{1}{{c + 1}} =  dfrac{{x + y + z}}{{ax + by + cz}}  end{array}$   Lại c&oacute; :   $ left {  begin{array}{l} x = by + cz   y = ax + cy   z = ax + by  end{array}  right.  Rightarrow x + y + z = 2 left( {ax + by + cz}  right)$   $  Rightarrow  dfrac{1}{{a + 1}} +  dfrac{1}{{b + 1}} +  dfrac{1}{{c + 1}} =  dfrac{{x + y + z}}{{ax + by + cz}} =  dfrac{{2 left( {ax + by + cz}  right)}}{{ax + by + cz}} = 2$