Solution. From the first equation of the system
2 x 1 − x 2 = 2 2x_1-x_2=2 2 x 1 − x 2 = 2 Divide through by 2 get
x 1 − 0.5 x 2 = 1. x_1 -0.5x_2=1. x 1 − 0.5 x 2 = 1. We write the system of equations as
( 1 − 0.5 0 1 2 − 1 0 1 3 ) ( x 1 x 2 x 3 ) = ( 1 1 3 ) . \begin{pmatrix}
1 & -0.5 & 0 \\
1 & 2 & -1 \\
0 & 1 & 3
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} = \begin{pmatrix}
1 \\
1 \\
3
\end{pmatrix} . ⎝ ⎛ 1 1 0 − 0.5 2 1 0 − 1 3 ⎠ ⎞ ⎝ ⎛ x 1 x 2 x 3 ⎠ ⎞ = ⎝ ⎛ 1 1 3 ⎠ ⎞ .
From the second equation of the system
x 1 + 2 x 2 − x 3 = 1 x_1 +2x_2-x_3=1 x 1 + 2 x 2 − x 3 = 1 Replacing x_1 -0.5*x_2=1 get
1 + 2.5 x 2 − x 3 = 1 → 2.5 x 2 − x 3 = 0 1 +2.5x_2-x_3=1 \to 2.5x_2-x_3=0 1 + 2.5 x 2 − x 3 = 1 → 2.5 x 2 − x 3 = 0 Divide through by 2.5 get
x 2 − 2 5 x 3 = 0. x_2-\frac{2}{5}x_3=0. x 2 − 5 2 x 3 = 0. We write the system of equations as
( 1 − 0.5 0 0 1 − 2 5 0 1 3 ) ( x 1 x 2 x 3 ) = ( 1 0 3 ) . \begin{pmatrix}
1 & -0.5 & 0 \\
0 & 1 & -\frac{2}{5} \\
0 & 1 & 3
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} = \begin{pmatrix}
1 \\
0 \\
3
\end{pmatrix} . ⎝ ⎛ 1 0 0 − 0.5 1 1 0 − 5 2 3 ⎠ ⎞ ⎝ ⎛ x 1 x 2 x 3 ⎠ ⎞ = ⎝ ⎛ 1 0 3 ⎠ ⎞ . From the third equation of the system
x 2 + 3 x 3 = 3 x_2+3x_3=3 x 2 + 3 x 3 = 3 Replacing x 2 − 2 5 x 3 = 0 x_2-\frac{2}{5}x_3=0 x 2 − 5 2 x 3 = 0 get
17 5 x 3 = 3 \frac {17}{5} x_3=3 5 17 x 3 = 3 Divide through by 17 5 \frac {17}{5} 5 17 get
x 3 = 15 17 x_3= \frac{15}{17} x 3 = 17 15 We write the system of equations as
( 1 − 0.5 0 0 1 − 2 5 0 0 1 ) ( x 1 x 2 x 3 ) = ( 1 0 15 17 ) . \begin{pmatrix}
1 & -0.5 & 0 \\
0 & 1 & -\frac{2}{5} \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} = \begin{pmatrix}
1 \\
0 \\
\frac {15}{17}
\end{pmatrix} . ⎝ ⎛ 1 0 0 − 0.5 1 0 0 − 5 2 1 ⎠ ⎞ ⎝ ⎛ x 1 x 2 x 3 ⎠ ⎞ = ⎝ ⎛ 1 0 17 15 ⎠ ⎞ . Let us find the roots of the equations of the system
x 3 = 15 17 x_3= \frac{15}{17} x 3 = 17 15
x 2 = 0 + 2 5 x 3 = 2 5 × 15 17 = 6 17 x_2=0+\frac{2}{5}x_3 = \frac{2}{5}\times \frac{15}{17}= \frac{6}{17} x 2 = 0 + 5 2 x 3 = 5 2 × 17 15 = 17 6
x 1 = 1 2 x 2 + 1 = 1 2 × 6 17 + 1 = 3 17 + 1 = 20 17 x_1 = \frac{1}{2}x_2+1=\frac{1}{2} \times \frac{6}{17} +1=\frac{3}{17} + 1 = \frac{20}{17} x 1 = 2 1 x 2 + 1 = 2 1 × 17 6 + 1 = 17 3 + 1 = 17 20
Answer. x 1 = 20 17 x_1 = \frac{20}{17} x 1 = 17 20 ; x 2 = 6 17 x_2= \frac{6}{17} x 2 = 17 6 ; x 3 = 15 17 x_3= \frac{15}{17} x 3 = 17 15 .
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