Problem 6 · AMC 8 Stretch
Core
Algebra & Patterns
add-and-subtract-the-equationsuse-symmetry
Find the pair of numbers \((x,y)\) that makes both equations true: \(123x+321y=345\) and \(321x+123y=543\).
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Answer: (x,y)=(3/2, 1/2)
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Hint 1 of 4
Don't panic at the big numbers. Notice the coefficients are mirror images: 123 and 321 just swap places. That's a hint to add and subtract the equations instead of substituting.
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Hint 2 of 4
Add the two equations. The \(x\)-coefficient becomes \(123+321=444\) and so does the \(y\)-coefficient, giving \(444x+444y=888\). Divide by 444.
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Hint 3 of 4
Subtract the first equation from the second. The \(x\)-coefficient becomes \(321-123=198\) and the \(y\)-coefficient becomes \(-198\), giving \(198x-198y=198\). Divide by 198.
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Approach: Use the mirror-image symmetry: add and subtract the equations
- The coefficients are mirror images, so add and subtract.
- Add the equations: \((123+321)x+(321+123)y=345+543\Rightarrow 444x+444y=888\Rightarrow x+y=2\).
- Subtract the first from the second: \((321-123)x+(123-321)y=543-345\Rightarrow 198x-198y=198\Rightarrow x-y=1\).
- Solve \(x+y=2\) and \(x-y=1\): adding gives \(2x=3\), so \(x=\tfrac32\), and then \(y=\tfrac12\). So \((x,y)=\left(\tfrac32,\tfrac12\right)\).
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