Finite Math Examples

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 1 & - 1 & - 1 & 0 & - 1 & 0 & 0 \\ - 0.07 & - 0.02 & - 0.13 & 0 & 0 & 1 & 0 \end{array}]$

Step 1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 1.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 1 - 1 & - 1 - 1 & - 1 - 1 & 0 - 1 & - 1 - 0 & 0 - 0 & 0 - 70000 \\ - 0.07 & - 0.02 & - 0.13 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & - 2 & - 2 & - 1 & - 1 & 0 & - 70000 \\ - 0.07 & - 0.02 & - 0.13 & 0 & 0 & 1 & 0 \end{array}]$

Step 2

Perform the row operation $R_{3} = R_{3} + 0.07 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{3} = R_{3} + 0.07 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & - 2 & - 2 & - 1 & - 1 & 0 & - 70000 \\ - 0.07 + 0.07 \cdot 1 & - 0.02 + 0.07 \cdot 1 & - 0.13 + 0.07 \cdot 1 & 0 + 0.07 \cdot 1 & 0 + 0.07 \cdot 0 & 1 + 0.07 \cdot 0 & 0 + 0.07 \cdot 70000 \end{array}]$

Step 2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & - 2 & - 2 & - 1 & - 1 & 0 & - 70000 \\ 0 & 0.05 & - 0.06 & 0.07 & 0 & 1 & 4900 \end{array}]$

Step 3

Multiply each element of $R_{2}$ by $- \frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

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Step 3.1

Multiply each element of $R_{2}$ by $- \frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 1 & - \frac{1}{2} \cdot - 1 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 70000 \\ 0 & 0.05 & - 0.06 & 0.07 & 0 & 1 & 4900 \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & 1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 35000 \\ 0 & 0.05 & - 0.06 & 0.07 & 0 & 1 & 4900 \end{array}]$

Step 4

Perform the row operation $R_{3} = R_{3} - 0.05 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.1

Perform the row operation $R_{3} = R_{3} - 0.05 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & 1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 35000 \\ 0 - 0.05 \cdot 0 & 0.05 - 0.05 \cdot 1 & - 0.06 - 0.05 \cdot 1 & 0.07 - 0.05 (\frac{1}{2}) & 0 - 0.05 (\frac{1}{2}) & 1 - 0.05 \cdot 0 & 4900 - 0.05 \cdot 35000 \end{array}]$

Step 4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & 1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 35000 \\ 0 & 0 & - 0.11 & 0.045 & - \frac{1}{40} & 1 & 3150 \end{array}]$

Step 5

Multiply each element of $R_{3}$ by $\frac{1}{- 0.11}$ to make the entry at $3, 3$ a $1$ .

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Step 5.1

Multiply each element of $R_{3}$ by $\frac{1}{- 0.11}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & 1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 35000 \\ \frac{0}{- 0.11} & \frac{0}{- 0.11} & \frac{- 0.11}{- 0.11} & \frac{0.045}{- 0.11} & \frac{- \frac{1}{40}}{- 0.11} & \frac{1}{- 0.11} & \frac{3150}{- 0.11} \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & 1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 35000 \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$

Step 6

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 6.1

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 - 0 & 1 - 0 & 1 - 1 & \frac{1}{2} + 0.4\overline{09} & \frac{1}{2} - 0.2\overline{27} & 0 + 9.\overline{09} & 35000 + 28636.\overline{36} \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$

Step 6.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 0 & 0 & 70000 \\ 0 & 1 & 0 & 0.\overline{90} & 0.\overline{27} & 9.\overline{09} & 63636.\overline{36} \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$

Step 7

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 7.1

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 1 + 0.4\overline{09} & 0 - 0.2\overline{27} & 0 + 9.\overline{09} & 70000 + 28636.\overline{36} \\ 0 & 1 & 0 & 0.\overline{90} & 0.\overline{27} & 9.\overline{09} & 63636.\overline{36} \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$

Step 7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 1.4\overline{09} & - 0.2\overline{27} & 9.\overline{09} & 98636.\overline{36} \\ 0 & 1 & 0 & 0.\overline{90} & 0.\overline{27} & 9.\overline{09} & 63636.\overline{36} \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$

Step 8

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 8.1

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 1.4\overline{09} - 0.\overline{90} & - 0.2\overline{27} - 0.\overline{27} & 9.\overline{09} - 9.\overline{09} & 98636.\overline{36} - 63636.\overline{36} \\ 0 & 1 & 0 & 0.\overline{90} & 0.\overline{27} & 9.\overline{09} & 63636.\overline{36} \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$

Step 8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0.5 & - 0.5 & 0 & 35000 \\ 0 & 1 & 0 & 0.\overline{90} & 0.\overline{27} & 9.\overline{09} & 63636.\overline{36} \\ 0 & 0 & 1 & - 0.4\overline{09} & 0.2\overline{27} & - 9.\overline{09} & - 28636.\overline{36} \end{array}]$