Precalculus Examples

$x - y + z = 6$ , $5 x + y - z = 0$

Step 1

Write the system as a matrix.

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

Step 2

Find the reduced row echelon form.

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 6 \\ 0 & 6 & - 6 & - 30 \end{array}]$

Step 2.2

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Step 2.2.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 1 & 6 \\ \frac{0}{6} & \frac{6}{6} & \frac{- 6}{6} & \frac{- 30}{6} \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify $R_{1}$ .

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

Step 3

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

$x = 1$

$y - z = - 5$

Step 4

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

$(1, - 5 + z, z)$