Question

Giải các hệ phương trình :

a) \(\left\{{}\begin{matrix}5x+3y=-7\\2x-4y=6\end{matrix}\right.\)

b) \(\left\{{}\begin{matrix}7x+14y=17\\2x+4y=5\end{matrix}\right.\)

c) \(\left\{{}\begin{matrix}\dfrac{2}{5}x+\dfrac{3}{7}y=\dfrac{1}{3}\\\dfrac{5}{3}x-\dfrac{5}{7}y=\dfrac{2}{3}\end{matrix}\right.\)

d) \(\left\{{}\begin{matrix}-0,2x+0,5y=1,7\\0,3x+0,4y=0,9\end{matrix}\right.\)

Answer 1

a) \(\left\{{}\begin{matrix}5x+3y=-7\\2x-4y=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5x+3y=-7\\x-2y=3\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}5x+3y=-7\\x=3+2y\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}5.\left(3+2y\right)+3y=-7\\x=3+2y\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}13y=-22\\x=3+2y\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-22}{13}\\x=3+2.\dfrac{-22}{13}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-22}{13}\\x=\dfrac{-5}{13}\end{matrix}\right.\)
Vậy hệ phương trình có nghiệm là: \(\left\{{}\begin{matrix}y=\dfrac{-22}{13}\\x=\dfrac{-5}{13}\end{matrix}\right.\).

Answer 2

b)\(\left\{{}\begin{matrix}7x+14y=17\\2x+4y=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}14x+28y=34\\14x+28y=35\end{matrix}\right.\) (vô nghiệm)
Vậy hệ phương trình vô nghiệm.

Answer 3

c) \(\left\{{}\begin{matrix}\dfrac{2}{5}x+\dfrac{3}{7}y=\dfrac{1}{3}\\\dfrac{5}{3}x-\dfrac{5}{7}=\dfrac{2}{3}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2}{5}x+\dfrac{3}{7}y=\dfrac{1}{3}\\x-\dfrac{3}{7}y=\dfrac{2}{5}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2}{5}x+\dfrac{3}{7}y=\dfrac{1}{3}\\x=\dfrac{3}{7}y+\dfrac{2}{5}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2}{5}.\left(\dfrac{3}{7}y+\dfrac{2}{5}\right)+\dfrac{3}{7}y=\dfrac{1}{3}\\x=\dfrac{3}{7}y+\dfrac{2}{5}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{3}{5}y=\dfrac{13}{75}\\x=\dfrac{3}{7}y+\dfrac{2}{5}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{13}{45}\\x=\dfrac{11}{21}\end{matrix}\right.\).
Vậy hệ phương trình có nghiệm là: \(\left\{{}\begin{matrix}y=\dfrac{13}{45}\\x=\dfrac{11}{21}\end{matrix}\right.\).

Answer 4

d) \(\left\{{}\begin{matrix}-0,2x+0,5y=1,7\\0,3x+0,4y=0,9\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}-2x+5y=17\\3x+4y=9\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}-6x+15y=51\\6x+8y=18\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}23y=69\\x=\dfrac{18-8y}{6}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=3\\x=-1\end{matrix}\right.\)
Vậy hệ phương trình có nghiệm: \(\left\{{}\begin{matrix}y=3\\x=-1\end{matrix}\right.\).