Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.2 AIME Find the answer United States

Problem:

The real numbers $x, y, z, w$ satisfy

$\begin{aligned} 2x + y + z + w &= 1 \\ x + 3y + z + w &= 2 \\ x + y + 4z + w &= 3 \\ x + y + z + 5w &= 25. \end{aligned}$

Find the value of $w$ .

Your answer

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

Solution

Solution:

$11/2$ . Multiplying the four equations by $12, 6, 4, 3$ respectively, we get

$\begin{aligned} 24x + 12y + 12z + 12w &= 12 \\ 6x + 18y + 6z + 6w &= 12 \\ 4x + 4y + 16z + 4w &= 12 \\ 3x + 3y + 3z + 15w &= 75 \end{aligned}$

Adding these yields $37x + 37y + 37z + 37w = 111$ , or $x + y + z + w = 3$ . Subtract this from the fourth given equation to obtain $4w = 22$ , or $w = 11/2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.