Finite Math Examples

$x - y = 2$ , $x - z = 2$ , $6 x - 2 y - 3 z = 3$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 1 & 0 & - 1 & 2 \\ 6 & - 2 & - 3 & 3 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 1 - 1 & 0 + 1 & - 1 - 0 & 2 - 2 \\ 6 & - 2 & - 3 & 3 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 0 & 1 & - 1 & 0 \\ 6 & - 2 & - 3 & 3 \end{array}]$

Step 2.2

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 0 & 1 & - 1 & 0 \\ 6 - 6 \cdot 1 & - 2 - 6 \cdot - 1 & - 3 - 6 \cdot 0 & 3 - 6 \cdot 2 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

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

Step 2.3

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 2.3.1

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 0 & 1 & - 1 & 0 \\ 0 & 0 & 1 & - 9 \end{array}]$

Step 2.4

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

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Step 2.4.1

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

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0 - 9 \\ 0 & 0 & 1 & - 9 \end{array}]$

Step 2.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 2 \\ 0 & 1 & 0 & - 9 \\ 0 & 0 & 1 & - 9 \end{array}]$

Step 2.5

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

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Step 2.5.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 2 - 9 \\ 0 & 1 & 0 & - 9 \\ 0 & 0 & 1 & - 9 \end{array}]$

Step 2.5.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 7 \\ 0 & 1 & 0 & - 9 \\ 0 & 0 & 1 & - 9 \end{array}]$

Step 3

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

$x = - 7$

$y = - 9$

$z = - 9$

Step 4

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

$(- 7, - 9, - 9)$