# Chapter 1 - Systems of Linear Equations - 1.2 Gaussian Elimination and Gauss-Jordan Elimination - 1.2 Exercises: 32

$x_1=8$ $x_2=10$ $x_3=6$

#### Work Step by Step

Write down the augmented matrix of the system of linear equations. \begin{bmatrix} 2 & -1 & 3 & 24 \\ 0 & 2 & -1 & 14 \\ 7& -5 & 0 &6 & \\ \end{bmatrix} $\frac{-7}{2} R_1+R_3 \rightarrow R_3$ \begin{bmatrix} 2 & -1 & 3 & 24 \\ 0 & 2 & -1 & 14 \\ 0& -3/2 & -21/2 &-78 & \\ \end{bmatrix} $\frac{3}{4} R_2+R_3 \rightarrow R_3$ \begin{bmatrix} 2 & -1 & 3 & 24 \\ 0 & 2 & -1 & 14 \\ 0& 0 &-45/4 &-135/2 & \\ \end{bmatrix} $4R_3 \rightarrow R_3$, getting rid of the fractions \begin{bmatrix} 2 & -1 & 3 & 24 \\ 0 & 2 & -1 & 14 \\ 0& 0 & -45 &-270 & \\ \end{bmatrix} Backsubstitution will be used to obtain the solution: $x_3=-270/-45=6$ $2x_2-6=14 \Rightarrow x_2=20/2=10$ $2x_1-10+3*6=24 \Rightarrow 2x_1=16 \Rightarrow x_1=8$

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