Finite Math Examples

$[\begin{array}{c} - 28 & 5 & - 3 & 84 \\ 10 & - 7 & - 25 & - 30 \\ 5 & - 1 & 0 & - 15 \end{array}]$

Step 1

Multiply each element of $R_{1}$ by $- \frac{1}{28}$ to make the entry at $1, 1$ a $1$ .

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Step 1.1

Multiply each element of $R_{1}$ by $- \frac{1}{28}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{28} \cdot - 28 & - \frac{1}{28} \cdot 5 & - \frac{1}{28} \cdot - 3 & - \frac{1}{28} \cdot 84 \\ 10 & - 7 & - 25 & - 30 \\ 5 & - 1 & 0 & - 15 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 10 & - 7 & - 25 & - 30 \\ 5 & - 1 & 0 & - 15 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 10 - 10 \cdot 1 & - 7 - 10 (- \frac{5}{28}) & - 25 - 10 (\frac{3}{28}) & - 30 - 10 \cdot - 3 \\ 5 & - 1 & 0 & - 15 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 0 & - \frac{73}{14} & - \frac{365}{14} & 0 \\ 5 & - 1 & 0 & - 15 \end{array}]$

Step 3

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 0 & - \frac{73}{14} & - \frac{365}{14} & 0 \\ 5 - 5 \cdot 1 & - 1 - 5 (- \frac{5}{28}) & 0 - 5 (\frac{3}{28}) & - 15 - 5 \cdot - 3 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 0 & - \frac{73}{14} & - \frac{365}{14} & 0 \\ 0 & - \frac{3}{28} & - \frac{15}{28} & 0 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $- \frac{14}{73}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $- \frac{14}{73}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ - \frac{14}{73} \cdot 0 & - \frac{14}{73} (- \frac{73}{14}) & - \frac{14}{73} (- \frac{365}{14}) & - \frac{14}{73} \cdot 0 \\ 0 & - \frac{3}{28} & - \frac{15}{28} & 0 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 0 & 1 & 5 & 0 \\ 0 & - \frac{3}{28} & - \frac{15}{28} & 0 \end{array}]$

Step 5

Perform the row operation $R_{3} = R_{3} + \frac{3}{28} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{3} = R_{3} + \frac{3}{28} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 0 & 1 & 5 & 0 \\ 0 + \frac{3}{28} \cdot 0 & - \frac{3}{28} + \frac{3}{28} \cdot 1 & - \frac{15}{28} + \frac{3}{28} \cdot 5 & 0 + \frac{3}{28} \cdot 0 \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{28} & \frac{3}{28} & - 3 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 6

Perform the row operation $R_{1} = R_{1} + \frac{5}{28} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 6.1

Perform the row operation $R_{1} = R_{1} + \frac{5}{28} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{5}{28} \cdot 0 & - \frac{5}{28} + \frac{5}{28} \cdot 1 & \frac{3}{28} + \frac{5}{28} \cdot 5 & - 3 + \frac{5}{28} \cdot 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 3 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$