Question #232441

Use the Gauss-Jordan Elimination method to solve the system of linear equations.

xa + 3xb + xc = 4

2xa + 2xb + xc = -1

2xa + 3xb + xc = 3



1
Expert's answer
2021-09-03T07:35:28-0400
xa+3xb+xc=42xa+2xb+xc=12xa+3xb+xc=3\begin{matrix} x_a+3x_b+x_c=4 \\ \\ 2x_a+2x_b+x_c=-1 \\ \\ 2x_a+3x_b+x_c=3 \\ \end{matrix}

The augmented matrix is


A=(131422112313)A=\begin{pmatrix} 1 & 3 & 1 & & 4 \\ 2 & 2 & 1 & & -1 \\ 2 & 3 & 1 & & 3 \\ \end{pmatrix}

R2=R22R1R_2=R_2-2R_1


(131404192313)\begin{pmatrix} 1 & 3 & 1 & & 4 \\ 0 & -4 & -1 & & -9 \\ 2 & 3 & 1 & & 3 \\ \end{pmatrix}

R3=R32R1R_3=R_3-2R_1


(131404190315)\begin{pmatrix} 1 & 3 & 1 & & 4 \\ 0 & -4 & -1 & & -9 \\ 0 & -3 & -1 & & -5 \\ \end{pmatrix}

R2=R2/(4)R_2=R_2/(-4)


(1314011/49/40315)\begin{pmatrix} 1 & 3 & 1 & & 4 \\ 0 & 1 & 1/4 & & 9/4 \\ 0 & -3 & -1 & & -5 \\ \end{pmatrix}

R1=R13R2R_1=R_1-3R_2


(101/411/4011/49/40315)\begin{pmatrix} 1 & 0 & 1/4 & & -11/4 \\ 0 & 1 & 1/4 & & 9/4 \\ 0 & -3 & -1 & & -5 \\ \end{pmatrix}

R3=R3+3R2R_3=R_3+3R_2


(101/411/4011/49/4001/47/4)\begin{pmatrix} 1 & 0 & 1/4 & & -11/4 \\ 0 & 1 & 1/4 & & 9/4 \\ 0 & 0 & -1/4 & & 7/4 \\ \end{pmatrix}

R3=4R3R_3=-4R_3


(101/411/4011/49/40017)\begin{pmatrix} 1 & 0 & 1/4 & & -11/4 \\ 0 & 1 & 1/4 & & 9/4 \\ 0 & 0 & 1 & & -7\\ \end{pmatrix}

R1=R1R3/4R_1=R_1-R_3/4


(1001011/49/40017)\begin{pmatrix} 1 & 0 & 0 & & -1 \\ 0 & 1 & 1/4 & & 9/4 \\ 0 & 0 & 1 & & -7\\ \end{pmatrix}

R2=R2R3/4R_2=R_2-R_3/4


(100101040017)\begin{pmatrix} 1 & 0 & 0 & & -1 \\ 0 & 1 & 0 & & 4 \\ 0 & 0 & 1 & & -7\\ \end{pmatrix}

Then xa=1,xb=4,xc=7.x_a=-1, x_b=4, x_c=-7.


The solution is (1,4,7).(-1, 4, -7).



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