Finite Math Examples

$x + y + z = - 1$ , $2 x + 3 y + 2 z = 3$ , $2 x + y + 2 z = - 7$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 & - 1 \\ 2 & 3 & 2 & 3 \\ 2 & 1 & 2 & - 7 \end{array}]$

Step 2

Find the reduced row echelon form.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$[\begin{array}{c} 1 & 1 & 1 & - 1 \\ 2 - 2 \cdot 1 & 3 - 2 \cdot 1 & 2 - 2 \cdot 1 & 3 - 2 \cdot - 1 \\ 2 & 1 & 2 & - 7 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & - 1 \\ 0 & 1 & 0 & 5 \\ 2 & 1 & 2 & - 7 \end{array}]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$[\begin{array}{c} 1 & 1 & 1 & - 1 \\ 0 & 1 & 0 & 5 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 1 & 2 - 2 \cdot 1 & - 7 - 2 \cdot - 1 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$[\begin{array}{c} 1 & 1 & 1 & - 1 \\ 0 & 1 & 0 & 5 \\ 0 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & - 5 + 1 \cdot 5 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 6 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3

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

$x + z = - 6$

$y = 5$

$0 = 0$

Step 4

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

$(- 6 - z, 5, z)$