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------------------------------------------------ Parameterize the solutions to the following system of linear equations, and write your formulas in vector form.

1. [tex]x_1 + x_3 - 3x_4 = 2[/tex]
2. [tex]x_2 + x_4 = -2[/tex]
3. [tex]x_3 + 3x_4 = -1[/tex]
4. [tex]-3x_1 + 2x_2 + x_3 + 23x_4 = -14[/tex]

To parameterize the solutions to a system of linear equations and write the formulas in vector form, rearrange the equations into matrix form and use Gaussian elimination to solve for the variables. Each variable can be written in vector form, such as x₁ = a, x₂ = b, x₃ = c, x₄ = d, where a, b, c, and d are the constants determined through the elimination process.

Explanation:

To parameterize the solutions to the given system of linear equations, we can write the equations in matrix form and use Gaussian elimination to solve for the variables. The system of equations can be written as:

x₁ + x₃ - 3x₄ = 2x₂ + x₄ = -2x₃ + 3x₄ = -1 - 3x₁ + 2x₂ + x₃ + 23x₄ = -14

By rearranging the equations into matrix form, we can perform Gaussian elimination to find the solutions. Each variable can be written in vector form, such as:

x₁ = a

x₂ = b

x₃ = c

x₄ = d

where a, b, c, and d are the constants determined through the elimination process. This will give us the parameterized solutions to the system of equations in vector form.

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