Linear Algebra Examples

$x_{1} + 2 x_{2} - 4 x_{3} = 4$ $- 2 x_{1} - 4 x_{2} + 8 x_{3} = - 8$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 2 & - 4 & 4 \\ - 2 & - 4 & 8 & - 8 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

$[\begin{array}{c} 1 & 2 & - 4 & 4 \\ - 2 + 2 \cdot 1 & - 4 + 2 \cdot 2 & 8 + 2 \cdot - 4 & - 8 + 2 \cdot 4 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

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

Step 3

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

$x_{1} + 2 x_{2} - 4 x_{3} = 4$

$0 = 0$

Step 4

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

$(4 - 2 x_{2} + 4 x_{3}, x_{2}, x_{3})$