Algebra Examples

$[\begin{array}{c} 10 & 4 & 20 & 2.95 \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 1

Multiply each element of $R_{1}$ by $\frac{1}{10}$ to make the entry at $1, 1$ a $1$ .

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Step 1.1

Multiply each element of $R_{1}$ by $\frac{1}{10}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{10}{10} & \frac{4}{10} & \frac{20}{10} & \frac{2.95}{10} \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 7.5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 7.5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 7.5 - 7.5 \cdot 1 & 6 - 7.5 (\frac{2}{5}) & 15 - 7.5 \cdot 2 & 3.77 - 7.5 \cdot 0.295 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 3

Perform the row operation $R_{3} = R_{3} - 15 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{3} = R_{3} - 15 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ 15 - 15 \cdot 1 & 3 - 15 (\frac{2}{5}) & 40 - 15 \cdot 2 & 1.8 - 15 \cdot 0.295 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ \frac{0}{3} & \frac{3}{3} & \frac{0}{3} & \frac{1.5575}{3} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

Step 5

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

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Step 5.1

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

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 10 + 3 \cdot 0 & - 2.625 + 3 \cdot 0.5191\overline{6} \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 10 & - 1.0674\overline{9} \end{array}]$

Step 6

Multiply each element of $R_{3}$ by $\frac{1}{10}$ to make the entry at $3, 3$ a $1$ .

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Step 6.1

Multiply each element of $R_{3}$ by $\frac{1}{10}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ \frac{0}{10} & \frac{0}{10} & \frac{10}{10} & \frac{- 1.0674\overline{9}}{10} \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Step 7

Perform the row operation $R_{1} = R_{1} - 2 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 7.1

Perform the row operation $R_{1} = R_{1} - 2 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & \frac{2}{5} - 2 \cdot 0 & 2 - 2 \cdot 1 & 0.295 - 2 \cdot - 0.10675 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Step 7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 0 & 0.5085 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Step 8

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

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Step 8.1

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

$[\begin{array}{c} 1 - \frac{2}{5} \cdot 0 & \frac{2}{5} - \frac{2}{5} \cdot 1 & 0 - \frac{2}{5} \cdot 0 & 0.5085 - \frac{2}{5} \cdot 0.5191\overline{6} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Step 8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0.3008\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$