Finite Math Examples

$x + 2 y + z = - 4$ , $- 2 x - 3 y - z = 1$ , $2 x + 4 y + 2 z = - 8$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 2 & 1 & - 4 \\ - 2 & - 3 & - 1 & 1 \\ 2 & 4 & 2 & - 8 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 2 & 1 & - 4 \\ - 2 + 2 \cdot 1 & - 3 + 2 \cdot 2 & - 1 + 2 \cdot 1 & 1 + 2 \cdot - 4 \\ 2 & 4 & 2 & - 8 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & - 4 \\ 0 & 1 & 1 & - 7 \\ 2 & 4 & 2 & - 8 \end{array}]$

Step 2.2

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

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Step 2.2.1

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

$[\begin{array}{c} 1 & 2 & 1 & - 4 \\ 0 & 1 & 1 & - 7 \\ 2 - 2 \cdot 1 & 4 - 2 \cdot 2 & 2 - 2 \cdot 1 & - 8 - 2 \cdot - 4 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & - 4 \\ 0 & 1 & 1 & - 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.3

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

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Step 2.3.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 1 - 2 \cdot 1 & - 4 - 2 \cdot - 7 \\ 0 & 1 & 1 & - 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 1 & 10 \\ 0 & 1 & 1 & - 7 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3

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

$x - z = 10$

$y + z = - 7$

$0 = 0$

Step 4

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

$(10 + z, - 7 - z, z)$