Finite Math Examples

$x + 4 y + 2 z = 21$ , $- x - 3 y - z = - 17$ , $3 x + 24 y + 18 z = 111$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ - 1 & - 3 & - 1 & - 17 \\ 3 & 24 & 18 & 111 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ - 1 + 1 \cdot 1 & - 3 + 1 \cdot 4 & - 1 + 1 \cdot 2 & - 17 + 1 \cdot 21 \\ 3 & 24 & 18 & 111 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ 0 & 1 & 1 & 4 \\ 3 & 24 & 18 & 111 \end{array}]$

Step 2.2

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

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Step 2.2.1

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

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ 0 & 1 & 1 & 4 \\ 3 - 3 \cdot 1 & 24 - 3 \cdot 4 & 18 - 3 \cdot 2 & 111 - 3 \cdot 21 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ 0 & 1 & 1 & 4 \\ 0 & 12 & 12 & 48 \end{array}]$

Step 2.3

Perform the row operation $R_{3} = R_{3} - 12 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 2.3.1

Perform the row operation $R_{3} = R_{3} - 12 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ 0 & 1 & 1 & 4 \\ 0 - 12 \cdot 0 & 12 - 12 \cdot 1 & 12 - 12 \cdot 1 & 48 - 12 \cdot 4 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 4 & 2 & 21 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.4

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 2.4.1

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 4 \cdot 0 & 4 - 4 \cdot 1 & 2 - 4 \cdot 1 & 21 - 4 \cdot 4 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 5 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3

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

$x - 2 z = 5$

$y + z = 4$

$0 = 0$

Step 4

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

$(5 + 2 z, 4 - z, z)$