Linear Algebra Examples

$x + 3 y - 4 z = 2$ $3 x + 10 y - 12 z = 8$ $9 x + 28 y - 36 z = 20$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 3 & 10 & - 12 & 8 \\ 9 & 28 & - 36 & 20 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 3 - 3 \cdot 1 & 10 - 3 \cdot 3 & - 12 - 3 \cdot - 4 & 8 - 3 \cdot 2 \\ 9 & 28 & - 36 & 20 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 0 & 1 & 0 & 2 \\ 9 & 28 & - 36 & 20 \end{array}]$

Step 2.2

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 0 & 1 & 0 & 2 \\ 9 - 9 \cdot 1 & 28 - 9 \cdot 3 & - 36 - 9 \cdot - 4 & 20 - 9 \cdot 2 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 1 & 0 & 2 \end{array}]$

Step 2.3

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

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Step 2.3.1

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

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 0 & 1 & 0 & 2 \\ 0 - 0 & 1 - 1 & 0 - 0 & 2 - 2 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 4 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.4

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

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Step 2.4.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 4 - 3 \cdot 0 & 2 - 3 \cdot 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 4 & - 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3

Use the result matrix to declare the final solution to the system of equations.

$x - 4 z = - 4$

$y = 2$

$0 = 0$

Step 4

The solution is the set of ordered pairs that make the system true.

$(- 4 + 4 z, 2, z)$