Competition problem — Algebra — Maths Olympiad Prep

Library / /141 of 353

Algebra Difficulty 4.8 AIME Find the answer

Compute $\frac{x}{w}$ if $w \neq 0$ and $\frac{x+6 y-3 z}{-3 x+4 w}=\frac{-2 y+z}{x-w}=\frac{2}{3}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have $x+6 y-3 z=\frac{2}{3}(-3 x+4 w)$ and $-2 y+z=\frac{2}{3}(x-w)$ , so $\frac{x}{w}=\frac{(x+6 y-3 z)+3(-2 y+z)}{(-3 x+4 w)+3(x-w)}=\frac{\frac{2}{3}(-3 x+4 w)+3 \cdot \frac{2}{3}(x-w)}{(-3 x+4 w)+3(x-w)}=\frac{\frac{2}{3}[(-3 x+4 w)+3(x-w)]}{(-3 x+4 w)+3(x-w)}=\frac{2}{3}$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.