Cho 4x-3y / 5 = 5y-4z/3 = 3z-5x/4 và x-y+z= 2020. Tìm x,y,z.

0 bình luận về “Cho 4x-3y / 5 = 5y-4z/3 = 3z-5x/4 và x-y+z= 2020. Tìm x,y,z.”

1. Đáp án:

   $\text{ x = 1515 , y = 2020 , z = 2525 }$ 

   Giải thích các bước giải:

   $\dfrac{4x-3y}{5}=$ $\dfrac{5y-4z}{4}=$ $\dfrac{3z-5x}{4}$ 

   $=\dfrac{5(4x-3y+3(5y-4z)+4(3z-5x)}{5.5+4.4+3.3}$

   $=\dfrac{20x-15y+15y-12z+12z-20x}{50}=$ $\dfrac{0}{50}=0$

   $⇔\left[\begin{array}{ccc}\dfrac{4x-3y}{5}=0\\\dfrac{5y-4z}{5}=0\\\dfrac{3z-5x}{5}=0\end{array}\right]$ $⇒\left[\begin{array}{ccc}4x-3y=0\\5y-4z=0\\3z-5x=0\end{array}\right]$ $⇒\left[\begin{array}{ccc}\dfrac{x}{3}= \dfrac{y}{4}\\\dfrac{y}{4}= \dfrac{z}{5}\\\dfrac{z}{5}= \dfrac{x}{3}\end{array}\right]$

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

   $\dfrac{x}{3}=$ $\dfrac{y}{4}=$ $\dfrac{z}{5}=$ $\dfrac{x-y+z}{3-4+5}=$ $\dfrac{2020}{4}=505$

   $⇒\dfrac{x}{3}=505⇒x=3.505=1515$

   $⇒\dfrac{y}{4}=505⇒y=4.505=2020$

   $⇒\dfrac{z}{5}=505⇒z=5.505=2525$

   Bình luận

2. Đáp án:

   $\left\{\begin{matrix}x=505.3=1515& \\y=505.4=2020&\\z=505.5=2525& \end{matrix}\right.$ 

   Giải thích các bước giải:

   `(4x-3y)/5=(5y-4z)/3=(3z-5x)/4`

   `=> (20x-15y)/25=(15y-12z)/9=(12z-20x)/16` 

   `=(20x-15y+15y-12z+12z-20x)/(25+9+16)=0/50=0`

   `=>` $\left\{\begin{matrix}\dfrac{4x-3y}{5}=0& \\\dfrac{5y-4z}{3}=0&\\\dfrac{3z-5x}{4}=0& \end{matrix}\right.$ `=>` $\left\{\begin{matrix}4x-3y=0& \\5y-4z=0&\\3z-5x=0& \end{matrix}\right.$ `=>` $\left\{\begin{matrix}4x=3y& \\5y=4z&\\3z=5x& \end{matrix}\right.$

   `=>` $\left\{\begin{matrix}\dfrac{x}{3}=\dfrac{y}{4}& \\\dfrac{y}{4}=\dfrac{z}{5}&\\\dfrac{z}{5}=\dfrac{x}{3}& \end{matrix}\right.$  `=> x/3=y/4=z/5`

   `=(x-y+z)/(3-4+5)=2020/4=505`

   `=>` $\left\{\begin{matrix}x=505.3=1515& \\y=505.4=2020&\\z=505.5=2525& \end{matrix}\right.$ 

   Bình luận