Indeterminate System of Linear Equations
Consider the system of equations
y y y y y = 2 x 1 + x 2 = 3 x 2 + x 3 = 4 x 3 + x 4 = 5 x 4 + x 5 = 6 x 5 + x 1 .
If all of the variables are integers, what is the minimum positive integer value of
( i = 1 ∑ 5 x i ) − y ?
The answer is 88.
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3 solutions
Exactly how I did it
same method XD
The problem does not require the x's and y to be integers (only x1 +... + x5 - y is required to be an integer). Thus one could argue that the correct answer is 1, with y = 721/88.
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Thanks. I agree that the question is ambiguous. Those who answered 1 have been marked correct. I have edited the problem and the correct answer is 88.
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The problem states that the values are integers, uness I missed something. Ed Gray
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Edits were made 3 years ago. Previously, it just required the final answer (but not the individual variables) to be integers.
Solving the simultaneous linear equations gives: x 1 =265, x 2 =191, x 3 = 148, x 4 = 129, x 5 = 76, so sum of x i, 1<=i<=5 =809. calculating any equation gives y = 721. The difference is 88. Ed Gray
I look for Xk in term of y. Then sum it up and then subtract by y, i get 88/721 y. Since xk and y is integer, hence i can put y=721 and get 88.
But is there any faster way?
By a little calculation, I have x 1 = 7 2 1 2 6 5 y , x 2 = 7 2 1 1 9 1 y , x 3 = 7 2 1 1 4 8 y , x 4 = 7 2 1 1 2 9 y , and x 5 = 7 2 1 7 6 y .
So,
x 1 + x 2 + x 3 + x 4 + x 5 − y = 7 2 1 8 8 y
Hence, the minimum positive integer value is 8 8 .