Cho a/b = c/d. Chứng minh (2a - 3b)/(2a + 3b) = (2c - 3d)/(2c + 3d)

Cho a/b = c/d. Chứng minh (2a - 3b)/(2a + 3b) = (2c - 3d)/(2c + 3d)

Đề bài: Cho $\frac{\text{a}}{\text{b}}\text{ = }\frac{\text{c}}{\text{d}}$. Chứng minh $\frac{\text{2a - 3b}}{\text{2a + 3b}}\text{ = }\frac{\text{2c - 3d}}{\text{2c + 3d}}$

Lời giải:

Đặt $\frac{\text{a}}{\text{b}}\text{ = }\frac{\text{c}}{\text{d}}\text{ = k}\left(\text{k}\in {\text{N}}^{\text{*}}\right)$

$\Rightarrow \left\{\begin{array}{c}a=bk\\ c=dk\end{array}\right.\Rightarrow \frac{2bk-3b}{2bk+3b}=\frac{2dk-3d}{2dk+3d}$

Xét vế trái $\frac{\text{2a - 3b}}{\text{2a + 3b}}\text{ = }\frac{\text{2bk - 3b}}{\text{2bk + 3b}}\text{ = }\frac{\text{b}\left(\text{2k - 3}\right)}{\text{b}\left(\text{2k+3}\right)}\text{=}\frac{\text{2k-3}}{\text{2k + 3}}\left(\text{1}\right)$

Xét vế phải $\frac{\text{2c - 3d}}{\text{2c + 3d}}\text{ = }\frac{\text{2dk - 3d}}{\text{2dk + 3d}}\text{ = }\frac{\text{d}\left(\text{2k - 3}\right)}{\text{d}\left(\text{2k+3}\right)}\text{=}\frac{\text{2k-3}}{\text{2k + 3}}\left(\text{2}\right)$

Từ (1) và (2) ta có Đpcm

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