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Answer :
Final answer:
To find all basic solutions of the given system of linear equations, we can use the method of Gaussian elimination with row operations.
Explanation:
To find all basic solutions of the given system of linear equations, we can use the method of Gaussian elimination. Here are the steps:
- Write the augmented matrix of the system.
- Perform row operations to transform the matrix into row-echelon form.
- Solve for the basic solutions using back-substitution.
Applying Gaussian elimination to the given system, we get:
2 3 4 1 | 144
0 -5 -7 3 | -21
0 0 1 1 | 17
From the row-echelon form, we can see that x4 is a free variable. Let's assign a parameter t to x4. Then using the values obtained from the row-echelon form, we can express the other variables in terms of t:
x3 = 17 - t
x2 = -21/(-5) - (-7/(-5))(17 - t) = 3 + 7(17 - t) = 52 - 7t
x1 = (144 - 4(52 - 7t) - 3(17 - t))/2 = 7t - 12
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