Finite Math Examples

$x + 9 y + 2 z = - 5$ , $4 y - 8 z = 1$ , $- 1 y + 2 z = - \frac{1}{4}$

Step 1

Rewrite $- 1 y$ as $- y$ .

$x + 9 y + 2 z = - 5$

$4 y - 8 z = 1$

$- y + 2 z = - \frac{1}{4}$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 9 & 2 & - 5 \\ 0 & 4 & - 8 & 1 \\ 0 & - 1 & 2 & - \frac{1}{4} \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 9 & 2 & - 5 \\ \frac{0}{4} & \frac{4}{4} & \frac{- 8}{4} & \frac{1}{4} \\ 0 & - 1 & 2 & - \frac{1}{4} \end{array}]$

Step 3.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 9 & 2 & - 5 \\ 0 & 1 & - 2 & \frac{1}{4} \\ 0 & - 1 & 2 & - \frac{1}{4} \end{array}]$

Step 3.2

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

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Step 3.2.1

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

$[\begin{array}{c} 1 & 9 & 2 & - 5 \\ 0 & 1 & - 2 & \frac{1}{4} \\ 0 + 0 & - 1 + 1 \cdot 1 & 2 - 2 & - \frac{1}{4} + \frac{1}{4} \end{array}]$

Step 3.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 9 & 2 & - 5 \\ 0 & 1 & - 2 & \frac{1}{4} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3

Perform the row operation $R_{1} = R_{1} - 9 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.1

Perform the row operation $R_{1} = R_{1} - 9 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 9 \cdot 0 & 9 - 9 \cdot 1 & 2 - 9 \cdot - 2 & - 5 - 9 (\frac{1}{4}) \\ 0 & 1 & - 2 & \frac{1}{4} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 20 & - \frac{29}{4} \\ 0 & 1 & - 2 & \frac{1}{4} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4

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

$x + 20 z = - \frac{29}{4}$

$y - 2 z = \frac{1}{4}$

$0 = 0$

Step 5

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

$(- \frac{29}{4} - 20 z, \frac{1}{4} + 2 z, z)$