Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\{\begin{array}{c}3x_1+5x_2-8x_3+5x_4=-8\\ x_1+2x_2-3x_3+x_4=-7\\ 2x_1+3x_2-7x_3+3x_4=-11\\ 4x_1+8x_2-10x_3+7x_4=-10\end{array}$

Verified step by step guidance

1. Write the system of equations in matrix form as an augmented matrix \([A|\mathbf{b}]\), where \(A\) is the coefficient matrix and \(\mathbf{b}\) is the constants column vector.
2. Use Gaussian elimination to transform the augmented matrix into an upper triangular form by applying row operations: swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another.
3. Once the matrix is in upper triangular form, use back-substitution to solve for the variables starting from the last row and moving upwards.
4. Alternatively, use Gauss-Jordan elimination to reduce the augmented matrix to reduced row echelon form (RREF), where the coefficient matrix becomes the identity matrix.
5. From the RREF matrix, directly read off the solutions for the variables, as each variable corresponds to a leading 1 in the identity matrix.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying matrix methods.

Gaussian Elimination with Back-Substitution

Gaussian elimination is a method to solve systems by transforming the augmented matrix into an upper triangular form using row operations. Back-substitution then solves for variables starting from the last equation upward. This stepwise approach simplifies complex systems into manageable forms.

Gauss-Jordan Elimination

Gauss-Jordan elimination extends Gaussian elimination by reducing the matrix further into reduced row-echelon form, where each leading coefficient is 1 and is the only nonzero entry in its column. This method directly provides the solution without needing back-substitution.