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50977 - Gaussian Elimination

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Time limit 2000/4000/4000/4000 ms. Memory limit 65000/65000/65000/65000 Kb.
Prepared by Ibrahim Mesecan.

Gaussian Elimination

Gaussian elimination, also known as row reduction, is a method to solve linear equations. The method is named after famous mathematician Carl Friedrich Gauss. The equation ax1 + bx2 = c is called as linear equation where a, b and c are numerical coefficients and xi is called as unknown variable. The previous equation has two unknowns. And, the following equation has three unknowns ax1 + bx2 + cx3 = d. In order to solve a linear equation with n unknowns, you need to have n (distinct) linear equations. Note: The solution steps are explained in this document.

Question: You will be given n-linear equations with n-unknowns. Calculate the unknown variable values and show them.

Input specification
You will be given an integer in the beginning: the number of unknowns (n). In the following n lines, you will be given n integers where 0 ≤ n ≤ 50. The following n lines will give the result of n linear equations.

Output specification:
Show n floating point numbers with 3 digits precision

Sample Input
3
1 2 3
4 5 6
1 0 1
1
1
1
Sample Output
0 -1 1

Explanation: There are three linear equations given where the first equation is x + 2y +3z = 1. The coefficients are 0, -1 and 1. Then, the first equation means: (1 * 0) + (2 * -1) + (3 * 1) = 1



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