Problem 28 · 2017 Math Kangaroo
Stretch
Algebra & Patterns
casework
How big is \(x + y\), if \(|x| + x + y = 5\) and \(x + |y| - y = 10\) both hold true?
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Answer: A — 1
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Hint 1 of 2
The absolute values force casework on the signs of x and y.
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Hint 2 of 2
Test the sign cases; only one keeps both equations consistent.
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Approach: casework on the signs inside the absolute values
- If x >= 0 then |x| + x + y = 2x + y = 5; if y < 0 then x + |y| - y = x - 2y = 10.
- Solving 2x + y = 5 and x - 2y = 10 gives x = 4, y = -3, consistent with x >= 0 and y < 0.
- Therefore x + y = 1.
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