若abc=1,解方程:2ax/(ab+a+1)+2bx/(bc+b+1)+2cx/(ca+c+1)=1

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/09 06:13:15

若abc=1,解方程:2ax/(ab+a+1)+2bx/(bc+b+1)+2cx/(ca+c+1)=1
若abc=1,解方程:2ax/(ab+a+1)+2bx/(bc+b+1)+2cx/(ca+c+1)=1

若abc=1,解方程:2ax/(ab+a+1)+2bx/(bc+b+1)+2cx/(ca+c+1)=1
a/(ab+a+1) = ac/(abc+ac+c) = ac/(ac+c+1)
b/(bc+b+1) = b/(bc+b+abc) = 1/(ac+c+1)
所以
2ax/(ab+a+1)+2bx/(bc+b+1)+2cx/(ca+c+1)
=2x*[ac/(ac+c+1)+1/(ac+c+1)+c/(ca+c+1)] = 2x = 1
所以x=1/2

2a/(ab+a+1)=2abc/(abc*b+abc+bc)=2/(b+bc+1) (分子分母同乘以bc)
2c/(ca+c+1)=2abc*bc/((abc)^2+abc*bc+abc*c)=2bc/(b+bc+1) (分子分母同乘以ab^2c)
因此原式转为:2(b+bc+1)x/(b+bc+1)=1,=>x=1/2

2ax/(ab+a+1)+2bx/(bc+b+1)+2cx/(ca+c+1)
=2ax/(ab+a+abc)+2bx/(bc+b+1)+2cx/(ca+c+1)
=2x/(b+bc+1)+2bx/(bc+b+1)+2cx/(ca+c+1)
=2(1+b)x/(1/a+b+1)+2cx/(ca+c+abc)
=2(a+ab)x/(a+ab+1)+2x/(a+1+ab)
=2x=1
故x=1/2