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CAT 2024 Slot 3 — QA Question 17

ModulusEasy

The number of distinct integer solutions (x, y) of the equation |x + y| + |x - y| = 2, is

Answer & solution

Answer: 8

Solution

Medium

Use the identity ∣x+y∣+∣x−y∣=2max⁡(∣x∣,∣y∣)|x+y|+|x-y|=2\max(|x|,|y|). The equation then forces max⁡(∣x∣,∣y∣)=1\max(|x|,|y|)=1, a small square boundary; count the lattice points on it.

1

Simplify.

∣x+y∣+∣x−y∣=2max⁡(∣x∣,∣y∣)⇒ 2max⁡(∣x∣,∣y∣)=2(given RHS)⇒ max⁡(∣x∣,∣y∣)=1\begin{aligned} &|x+y|+|x-y|=2\max(|x|,|y|)\\ &\Rightarrow\ 2\max(|x|,|y|)=2\quad\text{(given RHS)}\\ &\Rightarrow\ \max(|x|,|y|)=1 \end{aligned}
2

Count integer points. We need integers with the larger of ∣x∣,∣y∣|x|,|y| equal to 11, i.e. x,y∈{−1,0,1}x,y\in\{-1,0,1\} but not both 00.

points in {−1,0,1}2=3×3=9⇒ exclude (0,0) since max⁡=0≠1⇒ count=9−1=8\begin{aligned} &\text{points in }\{-1,0,1\}^2=3\times 3=9\\ &\Rightarrow\ \text{exclude }(0,0)\text{ since }\max=0\neq1\\ &\Rightarrow\ \text{count}=9-1=8 \end{aligned}
88

The solution set is the boundary of the square max⁡(∣x∣,∣y∣)=1\max(|x|,|y|)=1: the four corners (±1,±1)(\pm1,\pm1) and four edge midpoints (±1,0),(0,±1)(\pm1,0),(0,\pm1) — exactly 88 lattice points.

CAT 2024 Slot 3 QA Q17: The number of distinct integer solutions (x, y) of the equation |x + y| + |x - y| = 2, is — Solution | TheCATExam