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Answer :
Final answer:
This problem involves finding the general solution of a system of linear equations in vector form. We start by writing the system in augmented matrix form. Then, we apply Gaussian elimination or Gauss-Jordan method to reduce the matrix into Reduced Row Echelon Form (RREF) and identify the free parameters.
Explanation:
This problem set falls under the category of linear algebra and specifically relates to finding the solution of a system of linear equations in vector form. Given the presented system of equations, we need to write it in matrix form first. Label the columns of the matrix from x₁ to x₅ and augment the matrix with the right-hand of each equation as follows:
[1 3 -3 2 -3 | -43]
[ -9 -10 10 -14 | 22]
[ 2 6 -10 21 -5 | 53]
The '∣' symbol separates the coefficients of x terms on the left side from the constants on the right side. The goal is to reduce this augmented matrix to reduced row echelon form (RREF) using Gaussian elimination or Gauss-Jordan method, after which the solution can be read off directly in terms of free variables. It is important to remember that the result will be the general solution and will likely contain one or more 'free' or 'parameter' variables. Also note that the format for writing the vector form will essentially list these solutions. An example could be: [x₁, x₂, x₃, x₄, x₅] = t*[a, b, c, d, e] + s*[f, g, h, i, j], where t,s are parameters.
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