Precalculus Examples

$3 x - 2 y + 2 z - w = 2$ , $4 x + y + z + 6 w = 8$ , $- 3 x + 2 y - 2 z + w = 5$ , $5 x + 3 z - 2 w = 1$

Step 1

Move variables to the left and constant terms to the right.

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Step 1.1

Move $2 z$ .

$3 x - 2 y - w + 2 z = 2$

$4 x + y + z + 6 w = 8$

$- 3 x + 2 y - 2 z + w = 5$

$5 x + 3 z - 2 w = 1$

Step 1.2

Move $- 2 y$ .

$3 x - w - 2 y + 2 z = 2$

$4 x + y + z + 6 w = 8$

$- 3 x + 2 y - 2 z + w = 5$

$5 x + 3 z - 2 w = 1$

Step 1.3

Reorder $3 x$ and $- w$ .

$- w + 3 x - 2 y + 2 z = 2$

$4 x + y + z + 6 w = 8$

$- 3 x + 2 y - 2 z + w = 5$

$5 x + 3 z - 2 w = 1$

Step 1.4

Move $z$ .

$- w + 3 x - 2 y + 2 z = 2$

$4 x + y + 6 w + z = 8$

$- 3 x + 2 y - 2 z + w = 5$

$5 x + 3 z - 2 w = 1$

Step 1.5

Move $y$ .

$- w + 3 x - 2 y + 2 z = 2$

$4 x + 6 w + y + z = 8$

$- 3 x + 2 y - 2 z + w = 5$

$5 x + 3 z - 2 w = 1$

Step 1.6

Reorder $4 x$ and $6 w$ .

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$- 3 x + 2 y - 2 z + w = 5$

$5 x + 3 z - 2 w = 1$

Step 1.7

Move $- 2 z$ .

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$- 3 x + 2 y + w - 2 z = 5$

$5 x + 3 z - 2 w = 1$

Step 1.8

Move $2 y$ .

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$- 3 x + w + 2 y - 2 z = 5$

$5 x + 3 z - 2 w = 1$

Step 1.9

Reorder $- 3 x$ and $w$ .

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$w - 3 x + 2 y - 2 z = 5$

$5 x + 3 z - 2 w = 1$

Step 1.10

Move $3 z$ .

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$w - 3 x + 2 y - 2 z = 5$

$5 x - 2 w + 3 z = 1$

Step 1.11

Reorder $5 x$ and $- 2 w$ .

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$w - 3 x + 2 y - 2 z = 5$

$- 2 w + 5 x + 3 z = 1$

$- w + 3 x - 2 y + 2 z = 2$

$6 w + 4 x + y + z = 8$

$w - 3 x + 2 y - 2 z = 5$

$- 2 w + 5 x + 3 z = 1$

Step 2

Write the system as a matrix.

$[\begin{array}{c} - 1 & 3 & - 2 & 2 & 2 \\ 6 & 4 & 1 & 1 & 8 \\ 1 & - 3 & 2 & - 2 & 5 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{1}$ by $- 1$ to make the entry at $1, 1$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{1}$ by $- 1$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - - 1 & - 1 \cdot 3 & - - 2 & - 1 \cdot 2 & - 1 \cdot 2 \\ 6 & 4 & 1 & 1 & 8 \\ 1 & - 3 & 2 & - 2 & 5 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 6 & 4 & 1 & 1 & 8 \\ 1 & - 3 & 2 & - 2 & 5 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 6 - 6 \cdot 1 & 4 - 6 \cdot - 3 & 1 - 6 \cdot 2 & 1 - 6 \cdot - 2 & 8 - 6 \cdot - 2 \\ 1 & - 3 & 2 & - 2 & 5 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 22 & - 11 & 13 & 20 \\ 1 & - 3 & 2 & - 2 & 5 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3.3

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

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Step 3.3.1

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

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 22 & - 11 & 13 & 20 \\ 1 - 1 & - 3 + 3 & 2 - 2 & - 2 + 2 & 5 + 2 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 22 & - 11 & 13 & 20 \\ 0 & 0 & 0 & 0 & 7 \\ - 2 & 5 & 0 & 3 & 1 \end{array}]$

Step 3.4

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

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Step 3.4.1

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

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 22 & - 11 & 13 & 20 \\ 0 & 0 & 0 & 0 & 7 \\ - 2 + 2 \cdot 1 & 5 + 2 \cdot - 3 & 0 + 2 \cdot 2 & 3 + 2 \cdot - 2 & 1 + 2 \cdot - 2 \end{array}]$

Step 3.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 22 & - 11 & 13 & 20 \\ 0 & 0 & 0 & 0 & 7 \\ 0 & - 1 & 4 & - 1 & - 3 \end{array}]$

Step 3.5

Multiply each element of $R_{2}$ by $\frac{1}{22}$ to make the entry at $2, 2$ a $1$ .

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Step 3.5.1

Multiply each element of $R_{2}$ by $\frac{1}{22}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ \frac{0}{22} & \frac{22}{22} & \frac{- 11}{22} & \frac{13}{22} & \frac{20}{22} \\ 0 & 0 & 0 & 0 & 7 \\ 0 & - 1 & 4 & - 1 & - 3 \end{array}]$

Step 3.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 0 & 0 & 7 \\ 0 & - 1 & 4 & - 1 & - 3 \end{array}]$

Step 3.6

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

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Step 3.6.1

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

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 0 & 0 & 7 \\ 0 + 0 & - 1 + 1 \cdot 1 & 4 - \frac{1}{2} & - 1 + \frac{13}{22} & - 3 + \frac{10}{11} \end{array}]$

Step 3.6.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 0 & 0 & 7 \\ 0 & 0 & \frac{7}{2} & - \frac{9}{22} & - \frac{23}{11} \end{array}]$

Step 3.7

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & \frac{7}{2} & - \frac{9}{22} & - \frac{23}{11} \\ 0 & 0 & 0 & 0 & 7 \end{array}]$

Step 3.8

Multiply each element of $R_{3}$ by $\frac{2}{7}$ to make the entry at $3, 3$ a $1$ .

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Step 3.8.1

Multiply each element of $R_{3}$ by $\frac{2}{7}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ \frac{2}{7} \cdot 0 & \frac{2}{7} \cdot 0 & \frac{2}{7} \cdot \frac{7}{2} & \frac{2}{7} (- \frac{9}{22}) & \frac{2}{7} (- \frac{23}{11}) \\ 0 & 0 & 0 & 0 & 7 \end{array}]$

Step 3.8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 1 & - \frac{9}{77} & - \frac{46}{77} \\ 0 & 0 & 0 & 0 & 7 \end{array}]$

Step 3.9

Multiply each element of $R_{4}$ by $\frac{1}{7}$ to make the entry at $4, 5$ a $1$ .

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Step 3.9.1

Multiply each element of $R_{4}$ by $\frac{1}{7}$ to make the entry at $4, 5$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 1 & - \frac{9}{77} & - \frac{46}{77} \\ \frac{0}{7} & \frac{0}{7} & \frac{0}{7} & \frac{0}{7} & \frac{7}{7} \end{array}]$

Step 3.9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 1 & - \frac{9}{77} & - \frac{46}{77} \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.10

Perform the row operation $R_{3} = R_{3} + \frac{46}{77} R_{4}$ to make the entry at $3, 5$ a $0$ .

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Step 3.10.1

Perform the row operation $R_{3} = R_{3} + \frac{46}{77} R_{4}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 + \frac{46}{77} \cdot 0 & 0 + \frac{46}{77} \cdot 0 & 1 + \frac{46}{77} \cdot 0 & - \frac{9}{77} + \frac{46}{77} \cdot 0 & - \frac{46}{77} + \frac{46}{77} \cdot 1 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.10.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & \frac{10}{11} \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.11

Perform the row operation $R_{2} = R_{2} - \frac{10}{11} R_{4}$ to make the entry at $2, 5$ a $0$ .

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Step 3.11.1

Perform the row operation $R_{2} = R_{2} - \frac{10}{11} R_{4}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 - \frac{10}{11} \cdot 0 & 1 - \frac{10}{11} \cdot 0 & - \frac{1}{2} - \frac{10}{11} \cdot 0 & \frac{13}{22} - \frac{10}{11} \cdot 0 & \frac{10}{11} - \frac{10}{11} \cdot 1 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.11.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & - 2 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.12

Perform the row operation $R_{1} = R_{1} + 2 R_{4}$ to make the entry at $1, 5$ a $0$ .

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Step 3.12.1

Perform the row operation $R_{1} = R_{1} + 2 R_{4}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 + 2 \cdot 0 & - 3 + 2 \cdot 0 & 2 + 2 \cdot 0 & - 2 + 2 \cdot 0 & - 2 + 2 \cdot 1 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.12.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & 0 \\ 0 & 1 & - \frac{1}{2} & \frac{13}{22} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.13

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

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Step 3.13.1

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

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & 0 \\ 0 + \frac{1}{2} \cdot 0 & 1 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & \frac{13}{22} + \frac{1}{2} (- \frac{9}{77}) & 0 + \frac{1}{2} \cdot 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.13.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 2 & 0 \\ 0 & 1 & 0 & \frac{41}{77} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.14

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

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Step 3.14.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & - 3 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 2 - 2 (- \frac{9}{77}) & 0 - 2 \cdot 0 \\ 0 & 1 & 0 & \frac{41}{77} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.14.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & 0 & - \frac{136}{77} & 0 \\ 0 & 1 & 0 & \frac{41}{77} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.15

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

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Step 3.15.1

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

$[\begin{array}{c} 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 & - \frac{136}{77} + 3 (\frac{41}{77}) & 0 + 3 \cdot 0 \\ 0 & 1 & 0 & \frac{41}{77} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 3.15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{13}{77} & 0 \\ 0 & 1 & 0 & \frac{41}{77} & 0 \\ 0 & 0 & 1 & - \frac{9}{77} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 4

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

$w - \frac{13}{77} z = 0$

$x + \frac{41}{77} z = 0$

$y - \frac{9}{77} z = 0$

$0 = 1$

Step 5

The system is inconsistent so there is no solution.

$No solution$