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Absolute Sum

Problem

For all real numbers, $|x|$ is defined as the absolute value of $x$; for example $|4.2| = 4.2$ and $|-7| = 7$.

Given that $x$ and $y$ are integer, how many different solutions does the equation $|x| + 2|y| = 100$ have?


Solution

If $|y| \gt 50$, $2|y| \gt 100$, and $|x|$ would need to be negative, so $-50 \le y \le 50$.

If $y = \pm 50$ then $2|y| = 100 \implies x = 0$; that is, one solution for $y = -50$ and one solution for $y = 50$.

But for $-49 \le y \le 49$ there will be two values of $x$ for each value of $y$; for example, if $y = -20, 2|y| = 40, |x| = 60 \implies x = \pm 60$.

Therefore from -49 to 49 there are 49 (negative) + 1 (zero) + 49 (positive) = 99 values of $y$, each of which has two solutions.

Hence there are $2 \times 99 + 2 = 200$ distinct solutions to the equation.

If $k$ is a positive integer (odd or even), investigate the number of solutions of the equation $|x| + 2|y| = k$.
What about the equation $|x| + 3|y| = k$?
If the set of solutions are plotted as points, what shape is made?
What if $x$ and $y$ are not restricted to the set of integers?
Investigate the graph $\dfrac{|x|}{a} + \dfrac{|y|}{b} = 1$.

Problem ID: 292 (03 Oct 2006)     Difficulty: 3 Star

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