7) vì \(\dfrac{x}{5}\)=\(\dfrac{y}{6}\)=\(\dfrac{z}{7}\)và x-y+z=36

Nên theo tính chất của dãy tỉ số bằng nhau ta có:

 \(\dfrac{x}{5}\)=\(\dfrac{y}{6}\)=\(\dfrac{z}{7}\)=\(\dfrac{x-y+z}{5-6+7}\)=\(\dfrac{36}{6}\)=6

 \(\Rightarrow\)x=6.5=30

     y=6.6=36

     z=6.7=42

vậy x=30,y=36,z=42

\(\dfrac{x}{3}=\dfrac{y-5}{7}=\dfrac{z+2}{3}\)

\(\Leftrightarrow\dfrac{x}{3}=\dfrac{2y-10}{14}=\dfrac{5z+10}{15}\)

\(x+2y=5z\Leftrightarrow x+2y-5z=0\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{x}{3}=\dfrac{2y-10}{14}=\dfrac{5z+10}{15}=\dfrac{x+2y-10-5z-10}{3+14-15}\)

\(=\dfrac{-20}{2}=-10\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=-30\\y=-65\\z=-32\end{matrix}\right.\)

Vậy...

Áp dụng tính chất dãy tỉ số bằng nhau, ta có:

\(\dfrac{2x-y}{5}=\dfrac{3x-2z}{15}=\dfrac{2x-y-3x+2z}{5-15}=\dfrac{2\left(x+z\right)-4y}{-10}=\dfrac{4y-4y}{-10}=0\)

Do đó:

\(2x-y=0\Rightarrow2x=y\Rightarrow x=\dfrac{y}{2}\)

\(3y-2z=0\Rightarrow3y=2z\Rightarrow\dfrac{y}{2}=\dfrac{z}{3}\)

Vậy \(x=\dfrac{y}{2}=\dfrac{z}{3}\)

Từ x + z = 2y ta có:

x – 2y + z = 0 hay 2x – 4y + 2z = 0 hay 2x – y – 3y + 2z = 0 hay 2x – y = 3y – 2z

Vậy nếu: \(\dfrac{2x-y}{5}=\dfrac{3y-2z}{15}\) thì: 2x – y = 3y – 2z = 0 ﴾vì 5 \(\ne\)15.﴿

Từ 2x – y = 0 suy ra: x = \(\dfrac{1}{2}y\)

Từ 3y – 2z = 0 và x + z = 2y. x + z + y – 2z = 0 hay \(\dfrac{1}{2}y\) + y – z = 0 hay \(\dfrac{3}{2}y\) ‐ z = 0 hay y = \(\dfrac{2}{3}z\) . suy ra: x = \(\dfrac{1}{3}z\) .

Vậy các giá trị x, y, z cần tìm là: {x = \(\dfrac{1}{3}z\) ; y = \(\dfrac{2}{3}z\) ; với z \(\in\) R } hoặc {x =\(\dfrac{1}{2}y\) ; z = \(\dfrac{3}{2}y\);với y \(\in\) R} hoặc { y = 2x; z = 3x ;với x \(\in\)R}

Ap dung tinh chat day ti so bang nhau, ta co:

\(\dfrac{2x-y}{5}=\dfrac{3y-2z}{15}=\dfrac{2x-y-3y+2z}{5-15}=\dfrac{2\left(x+z\right)-4y}{-10}=\dfrac{4y-4y}{-10}=0\)

Do do

\(2x-y=0\Rightarrow2x=y\Rightarrow x=\dfrac{y}{2}\)

\(3y-2z=0\Rightarrow3y=2z\Rightarrow\dfrac{y}{2}=\dfrac{z}{3}\)

Vay \(x=\dfrac{y}{2}=\dfrac{z}{3}\)

a) Ta có: \(\left|2x+5\right|+\left|2x-3\right|=8\)

\(\Rightarrow\left|2x+5\right|+\left|3-2x\right|=8\)

Nhận thấy \(\left[{}\begin{matrix}\left|2x+5\right|\ge2x+5\forall x\\\left|3-2x\right|\ge3-2x\forall x\end{matrix}\right.\)

\(\Rightarrow\left|2x+5\right|+\left|3-2x\right|\ge2x+5+3-2x\forall x\)

\(\Rightarrow\left|2x+5\right|+\left|3-2x\right|\ge8\)

Dấu \("="\) xảy ra khi \(\left[{}\begin{matrix}2x+5\ge0\\3-2x\ge0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}2x\ge-5\\2x\le3\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x\ge\dfrac{-5}{2}\\x\le\dfrac{3}{2}\end{matrix}\right.\) \(\Rightarrow\dfrac{-5}{2}\le x\le\dfrac{3}{2}\)

Vậy \(\dfrac{-5}{2}\le x\le\dfrac{3}{2}.\)