Đáp án:
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`a,`
Áp dụng tính chất dãy tỉ số bằng nhau có :
`x/3 = y/4 = z/6 = (x + y + z)/(3 + 4 + 6) = 52/13 = 4`
`↔` \(\left\{ \begin{array}{l}\dfrac{x}{3}=4\\ \dfrac{y}{4}=4\\ \dfrac{z}{6}=4\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=3×4\\y=4×4\\z=6×4\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=12\\y=16\\z=24\end{array} \right.\)
Vậy `(x;y;z) = (12;16;24)`
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`b,`
Đặt `(x - 1)/2 = (y+2)/3 = (z-3)/4 = k`
`↔` \(\left\{ \begin{array}{l}\dfrac{x-1}{2}=k\\ \dfrac{y+2}{3}=k\\ \dfrac{z-3}{4}=k\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x-1=2k\\y+2=3k\\z-3=4k\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=2k+1\\y=3k-2\\z=4k+3\end{array} \right.\) `(1)`
Có : `x - 2y + 3z = 46`
Thay `(1)` vào ta được :
`↔ 2k + 1 - 2 (3k - 2) + 3 (4k + 3) = 46`
`↔ 2k + 1 - 6k + 4 + 12k + 9 = 46`
`↔ (2k - 6k + 12k) + (1 + 4 + 9) = 46`
`↔ 8k + 14 = 46`
`↔ 8k = 46 -14`
`↔ 8k = 32`
`↔ k = 32 ÷ 8`
`↔ k = 4`
Với `k=4` thay vào `(1)` ta được :
`↔` \(\left\{ \begin{array}{l}x=2 × 4 +1 \\y=3×4-2\\z=4×4+3\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=8+1\\y=12-2\\z=16+3\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=9\\y=10\\z=19\end{array} \right.\)
Vậy `(x;y;z) = (9; 10;19)`
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`c,`
Có : `x/y = 7/10`
`↔ x/7 = y/10` `(1)`
Có : `y/z = 5/8`
`↔ y/5 = z/8`
`↔ y/10 = z/16` `(2)`
Từ `(1), (2)`
`↔ x/7 = y/10 = z/16`
`↔ (2x)/14 = y/10 = (3z)/48`
Áp dụng tính chất dãy tỉ số bằng nhau có :
`(2x)/14 = y/10 = (3z)/48 = (2x-y+3z)/(14 - 10 + 48) = 104/52 = 2`
`↔` \(\left\{ \begin{array}{l}\dfrac{2x}{14}=2\\ \dfrac{y}{10}=2\\ \dfrac{3z}{48}=2\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}2x=14×2\\y=10×2\\3z=48×2\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}2x=28\\y=20\\3z=96\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=28÷2\\y=20\\z=96÷3\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=14\\y=20\\z=32\end{array} \right.\)
Vậy `(x;y;z) = (14;20;32)`
