Question

Solve the system of equations by row-reduction. At each step, show clearly the symbol of the linear combinations that allow you to clear the entries below each pivot or of the operations that allow you to swap two rows or to scale a row.

1) 3x₂ - 5x₃ = 89
6x₁ + x₃ = 17
x₁ - x₂ + 8x₃ = -107
2) 4x₁ - x₂ + 3x₃ = 12
2x₁ + 9x₃ = -5
x₁ + 4x₂ + 6x₃ = -32

Answer 1

1) The solution of the system is

`\left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right`

2) The solution of the system is

`\left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right`

Explanation:

1) To solve the system of equations

`\left\begin{array}{ccccccc}&3x_2&-5x_3&=&89\\6x_1&&+x_3&=&17\\x_1&-x_2&+8x_3&=&-107\end{array}\right`

using the row reduction method you must:

Step 1: Write the augmented matrix of the system

`\left[ \begin{array}ccc 0 & 3 & -5 & 89 \\\\ 6 & 0 & 1 & 17 \\\\ 1 & -1 & 8 & -107 \end{array} \right]`

Step 2: Swap rows 1 and 2

`\left[ \begin{array}ccc 6 & 0 & 1 & 17 \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right]`

Step 3:
`\left(R_1=(R_1)/(6)\right)`

`\left[ \begin{array}ccc 1 & 0 & (1)/(6) & (17)/(6) \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right]`

Step 4:
`\left(R_3=R_3-R_1\right)`

`\left[ \begin{array}c 1 & 0 & (1)/(6) & (17)/(6) \\\\ 0 & 3 & -5 & 89 \\\\ 0 & -1 & (47)/(6) & - (659)/(6) \end{array} \right]`

Step 5:
`\left(R_2=(R_2)/(3)\right)`

`\left[ \begin{array}c 1 & 0 & (1)/(6) & (17)/(6) \\\\ 0 & 1 & - (5)/(3) & (89)/(3) \\\\ 0 & -1 & (47)/(6) & - (659)/(6) \end{array} \right]`

Step 6:
`\left(R_3=R_3+R_2\right)`

`\left[ \begin{array}ccc 1 & 0 & (1)/(6) & (17)/(6) \\\\ 0 & 1 & - (5)/(3) & (89)/(3) \\\\ 0 & 0 & (37)/(6) & - (481)/(6) \end{array} \right]`

Step 7:
`\left(R_3=\left((6)/(37)\right)R_3\right)`

`\left[ \begin{array}ccc 1 & 0 & (1)/(6) & (17)/(6) \\\\ 0 & 1 & - (5)/(3) & (89)/(3) \\\\ 0 & 0 & 1 & -13 \end{array} \right]`

Step 8:
`\left(R_1=R_1-\left((1)/(6)\right)R_3\right)`

`\left[ \begin{array}c 1 & 0 & 0 & 5 \\\\ 0 & 1 & - (5)/(3) & (89)/(3) \\\\ 0 & 0 & 1 & -13 \end{array} \right]`

Step 9:
`\left(R_2=R_2+\left((5)/(3)\right)R_3\right)`

`\left[ \begin{array}c 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right]`

Step 10: Rewrite the system using the row reduced matrix:

`\left[ \begin{array}ccc 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right] \rightarrow \left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right`

2) To solve the system of equations

`\left\begin{array}{ccccccc}4x_1&-x_2&+3x_3&=&12\\2x_1&&+9x_3&=&-5\\x_1&+4x_2&+6x_3&=&-32\end{array}\right`

using the row reduction method you must:

Step 1:

`\left[ \begin{array}c 4 & -1 & 3 & 12 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right]`

Step 2:
`\left(R_1=(R_1)/(4)\right)`

`\left[ \begin{array}ccc 1 & - (1)/(4) & (3)/(4) & 3 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right]`

Step 3:
`\left(R_2=R_2-\left(2\right)R_1\right)`

`\left[ \begin{array}c 1 & - (1)/(4) & (3)/(4) & 3 \\\\ 0 & (1)/(2) & (15)/(2) & -11 \\\\ 1 & 4 & 6 & -32 \end{array} \right]`

Step 4:
`\left(R_3=R_3-R_1\right)`

`\left[ \begin{array}ccc 1 & - (1)/(4) & (3)/(4) & 3 \\\\ 0 & (1)/(2) & (15)/(2) & -11 \\\\ 0 & (17)/(4) & (21)/(4) & -35 \end{array} \right]`

Step 5:
`\left(R_2=\left(2\right)R_2\right)`

`\left[ \begin{array}ccc 1 & - (1)/(4) & (3)/(4) & 3 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & (17)/(4) & (21)/(4) & -35 \end{array} \right]`

Step 6:
`\left(R_1=R_1+\left((1)/(4)\right)R_2\right)`

`\left[ \begin{array}{cccc} 1 & 0 & (9)/(2) & - (5)/(2) \\\\ 0 & 1 & 15 & -22 \\\\ 0 & (17)/(4) & (21)/(4) & -35 \end{array} \right]`

Step 7:
`\left(R_3=R_3-\left((17)/(4)\right)R_2\right)`

`\left[ \begin{array}ccc 1 & 0 & (9)/(2) & - (5)/(2) \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & - (117)/(2) & (117)/(2) \end{array} \right]`

Step 8:
`\left(R_3=\left(- (2)/(117)\right)R_3\right)`

`\left[ \begin{array}{cccc} 1 & 0 & (9)/(2) & - (5)/(2) \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right]`

Step 9:
`\left(R_1=R_1-\left((9)/(2)\right)R_3\right)`

`\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right]`

Step 10:
`\left(R_2=R_2-\left(15\right)R_3\right)`

`\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right]`

Step 11:

`\left[ \begin{array}ccc 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right]\rightarrow \left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right`