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Cho các số a,b,c>0 và \(\dfrac{a+b}{3}=\dfrac{b+c}{4}=\dfrac{c+a}{5}\)
Tính GTBT \(M=10a+b-7c+2017\)
Đặt \(\dfrac{a+b}{3}=\dfrac{b+c}{4}=\dfrac{c+a}{5}=k\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=3k\\b+c=4k\\c+a=5k\end{matrix}\right.\Rightarrow2\cdot\left(a+b+c\right)=12k\Rightarrow a+b+c=6k\)
\(\Rightarrow\left\{{}\begin{matrix}c=3k\left(1\right)\\a=2k\left(2\right)\\b=k\left(3\right)\end{matrix}\right.\)
Thay \(\left(1\right),\left(2\right),\left(3\right)\) vào BT ta có:
\(M=10\cdot2k+k-7\cdot3k+2017\)
\(M=20k+k-21k+2017\)
\(M=21k-21k+2017\)
\(M=2017\)
Vậy \(M=2017\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\dfrac{a+b}{3}=\dfrac{b+c}{4}=\dfrac{c+a}{5}=\dfrac{a+b+b+c+c+a}{3+4+5}=\dfrac{a+b+b+c}{3+4}=\dfrac{b+c+c+a}{4+5}=\dfrac{a+b+c+a}{3+5}=\dfrac{a+b+c}{6}=\dfrac{a+2b+c}{7}=\dfrac{b+2c+a}{9}=\dfrac{2a+b+c}{8}\)
Tiếp tục áp dụng tính chất dãy tỉ số bằng nhau:
\(\dfrac{a+b+c}{6}=\dfrac{a+2b+c}{7}=\dfrac{b+2c+a}{9}=\dfrac{2a+b+c}{8}=\dfrac{2a+b+c-a-b-c}{8-6}=\dfrac{a}{2}\left(1\right)\)
\(\dfrac{a+b+c}{6}=\dfrac{a+2b+c}{7}=\dfrac{b+2c+a}{9}=\dfrac{2a+b+c}{8}=\dfrac{a+2b+c-a-b-c}{7-6}=\dfrac{b}{1}\left(2\right)\)
\(\dfrac{a+b+c}{6}=\dfrac{a+2b+c}{7}=\dfrac{b+2c+a}{9}=\dfrac{2a+b+c}{8}=\dfrac{b+2c+a-a-b-c}{9-6}=\dfrac{c}{3}\left(3\right)\)
Từ (1);(2) và (3) ta có: \(\dfrac{a}{2}=\dfrac{b}{1}=\dfrac{c}{3}\)
Đặt: \(\dfrac{a}{2}=\dfrac{b}{1}=\dfrac{c}{3}=t\Leftrightarrow\left\{{}\begin{matrix}a=2t\\b=t\\c=3t\end{matrix}\right.\)
Thay vào \(M\) ta có: \(M=10a+b-7c+2017=20t+t-21t+2017=2017\)
\(\dfrac{a+b}{3}=\dfrac{b+c}{4}=\dfrac{c+a}{5}\\ \Rightarrow\left\{{}\begin{matrix}4a+4b=3b+3c\\5a+5b=3c+3a\\5b+5c=4c+4a\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}4a+b-3c=0\\4a-5b-c=0\\2a+5b-3c=0\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}4a+b-3c=0\\4a=5b+c\\3c=2a+5b\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}5b+c+b-3c=0\\4a+b-2a-5b=0\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}6b=2c\\2a=4b\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}c=3b\\a=2b\end{matrix}\right.\\ \Rightarrow M=20b+b-21b+2021=2021\)
Đặt \(\dfrac{a+b}{3}=\dfrac{b+c}{4}=\dfrac{c+a}{5}=k\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=3k\\b+c=4k\\c+a=5k\end{matrix}\right.\)
\(\Rightarrow2\left(a+b+c\right)=12k\)
\(\Rightarrow a+b+c=6k\) \(\Rightarrow\left\{{}\begin{matrix}a=2k\\b=k\\c=3k\end{matrix}\right.\)
Thay a = 2k , b = k , c= = 3k vào biểu thức M , ta có :
M = 10.2k + k - 7.3k + 2017 = 21k - 21k + 2017 = 2017