Precalculus Examples

$x + 3 y - 4 z - w = 12$ , $4 x + y + 2 z + 4 w = 6$ , $- 3 x - y + z - 3 w = - 10$ , $x - y - 3 z - 2 w = 0$

Step 1

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

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Step 1.1

Move $- 4 z$ .

$x + 3 y - w - 4 z = 12$

$4 x + y + 2 z + 4 w = 6$

$- 3 x - y + z - 3 w = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.2

Move $3 y$ .

$x - w + 3 y - 4 z = 12$

$4 x + y + 2 z + 4 w = 6$

$- 3 x - y + z - 3 w = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.3

Reorder $x$ and $- w$ .

$- w + x + 3 y - 4 z = 12$

$4 x + y + 2 z + 4 w = 6$

$- 3 x - y + z - 3 w = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.4

Move $2 z$ .

$- w + x + 3 y - 4 z = 12$

$4 x + y + 4 w + 2 z = 6$

$- 3 x - y + z - 3 w = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.5

Move $y$ .

$- w + x + 3 y - 4 z = 12$

$4 x + 4 w + y + 2 z = 6$

$- 3 x - y + z - 3 w = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.6

Reorder $4 x$ and $4 w$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 x - y + z - 3 w = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.7

Move $z$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 x - y - 3 w + z = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.8

Move $- y$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 x - 3 w - y + z = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.9

Reorder $- 3 x$ and $- 3 w$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 w - 3 x - y + z = - 10$

$x - y - 3 z - 2 w = 0$

Step 1.10

Move $- 3 z$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 w - 3 x - y + z = - 10$

$x - y - 2 w - 3 z = 0$

Step 1.11

Move $- y$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 w - 3 x - y + z = - 10$

$x - 2 w - y - 3 z = 0$

Step 1.12

Reorder $x$ and $- 2 w$ .

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 w - 3 x - y + z = - 10$

$- 2 w + x - y - 3 z = 0$

$- w + x + 3 y - 4 z = 12$

$4 w + 4 x + y + 2 z = 6$

$- 3 w - 3 x - y + z = - 10$

$- 2 w + x - y - 3 z = 0$

Step 2

Write the system as a matrix.

$[\begin{array}{c} - 1 & 1 & 3 & - 4 & 12 \\ 4 & 4 & 1 & 2 & 6 \\ - 3 & - 3 & - 1 & 1 & - 10 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

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 $1$ .

$[\begin{array}{c} - - 1 & - 1 \cdot 1 & - 1 \cdot 3 & - - 4 & - 1 \cdot 12 \\ 4 & 4 & 1 & 2 & 6 \\ - 3 & - 3 & - 1 & 1 & - 10 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 4 & 4 & 1 & 2 & 6 \\ - 3 & - 3 & - 1 & 1 & - 10 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

Step 3.2

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

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Step 3.2.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 4 - 4 \cdot 1 & 4 - 4 \cdot - 1 & 1 - 4 \cdot - 3 & 2 - 4 \cdot 4 & 6 - 4 \cdot - 12 \\ - 3 & - 3 & - 1 & 1 & - 10 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 8 & 13 & - 14 & 54 \\ - 3 & - 3 & - 1 & 1 & - 10 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

Step 3.3

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

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Step 3.3.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 8 & 13 & - 14 & 54 \\ - 3 + 3 \cdot 1 & - 3 + 3 \cdot - 1 & - 1 + 3 \cdot - 3 & 1 + 3 \cdot 4 & - 10 + 3 \cdot - 12 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

Step 3.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 8 & 13 & - 14 & 54 \\ 0 & - 6 & - 10 & 13 & - 46 \\ - 2 & 1 & - 1 & - 3 & 0 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 8 & 13 & - 14 & 54 \\ 0 & - 6 & - 10 & 13 & - 46 \\ - 2 + 2 \cdot 1 & 1 + 2 \cdot - 1 & - 1 + 2 \cdot - 3 & - 3 + 2 \cdot 4 & 0 + 2 \cdot - 12 \end{array}]$

Step 3.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 8 & 13 & - 14 & 54 \\ 0 & - 6 & - 10 & 13 & - 46 \\ 0 & - 1 & - 7 & 5 & - 24 \end{array}]$

Step 3.5

Multiply each element of $R_{2}$ by $\frac{1}{8}$ to make the entry at $2, 2$ a $1$ .

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Step 3.5.1

Multiply each element of $R_{2}$ by $\frac{1}{8}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ \frac{0}{8} & \frac{8}{8} & \frac{13}{8} & \frac{- 14}{8} & \frac{54}{8} \\ 0 & - 6 & - 10 & 13 & - 46 \\ 0 & - 1 & - 7 & 5 & - 24 \end{array}]$

Step 3.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & - 6 & - 10 & 13 & - 46 \\ 0 & - 1 & - 7 & 5 & - 24 \end{array}]$

Step 3.6

Perform the row operation $R_{3} = R_{3} + 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 3.6.1

Perform the row operation $R_{3} = R_{3} + 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & - 10 + 6 (\frac{13}{8}) & 13 + 6 (- \frac{7}{4}) & - 46 + 6 (\frac{27}{4}) \\ 0 & - 1 & - 7 & 5 & - 24 \end{array}]$

Step 3.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & - \frac{1}{4} & \frac{5}{2} & - \frac{11}{2} \\ 0 & - 1 & - 7 & 5 & - 24 \end{array}]$

Step 3.7

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

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Step 3.7.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & - \frac{1}{4} & \frac{5}{2} & - \frac{11}{2} \\ 0 + 0 & - 1 + 1 \cdot 1 & - 7 + \frac{13}{8} & 5 - \frac{7}{4} & - 24 + \frac{27}{4} \end{array}]$

Step 3.7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & - \frac{1}{4} & \frac{5}{2} & - \frac{11}{2} \\ 0 & 0 & - \frac{43}{8} & \frac{13}{4} & - \frac{69}{4} \end{array}]$

Step 3.8

Multiply each element of $R_{3}$ by $- 4$ to make the entry at $3, 3$ a $1$ .

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Step 3.8.1

Multiply each element of $R_{3}$ by $- 4$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ - 4 \cdot 0 & - 4 \cdot 0 & - 4 (- \frac{1}{4}) & - 4 (\frac{5}{2}) & - 4 (- \frac{11}{2}) \\ 0 & 0 & - \frac{43}{8} & \frac{13}{4} & - \frac{69}{4} \end{array}]$

Step 3.8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & 1 & - 10 & 22 \\ 0 & 0 & - \frac{43}{8} & \frac{13}{4} & - \frac{69}{4} \end{array}]$

Step 3.9

Perform the row operation $R_{4} = R_{4} + \frac{43}{8} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 3.9.1

Perform the row operation $R_{4} = R_{4} + \frac{43}{8} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & 1 & - 10 & 22 \\ 0 + \frac{43}{8} \cdot 0 & 0 + \frac{43}{8} \cdot 0 & - \frac{43}{8} + \frac{43}{8} \cdot 1 & \frac{13}{4} + \frac{43}{8} \cdot - 10 & - \frac{69}{4} + \frac{43}{8} \cdot 22 \end{array}]$

Step 3.9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & 1 & - 10 & 22 \\ 0 & 0 & 0 & - \frac{101}{2} & 101 \end{array}]$

Step 3.10

Multiply each element of $R_{4}$ by $- \frac{2}{101}$ to make the entry at $4, 4$ a $1$ .

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Step 3.10.1

Multiply each element of $R_{4}$ by $- \frac{2}{101}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & 1 & - 10 & 22 \\ - \frac{2}{101} \cdot 0 & - \frac{2}{101} \cdot 0 & - \frac{2}{101} \cdot 0 & - \frac{2}{101} (- \frac{101}{2}) & - \frac{2}{101} \cdot 101 \end{array}]$

Step 3.10.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & 1 & - 10 & 22 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.11

Perform the row operation $R_{3} = R_{3} + 10 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 3.11.1

Perform the row operation $R_{3} = R_{3} + 10 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 + 10 \cdot 0 & 0 + 10 \cdot 0 & 1 + 10 \cdot 0 & - 10 + 10 \cdot 1 & 22 + 10 \cdot - 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.11.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & - \frac{7}{4} & \frac{27}{4} \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.12

Perform the row operation $R_{2} = R_{2} + \frac{7}{4} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 3.12.1

Perform the row operation $R_{2} = R_{2} + \frac{7}{4} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 + \frac{7}{4} \cdot 0 & 1 + \frac{7}{4} \cdot 0 & \frac{13}{8} + \frac{7}{4} \cdot 0 & - \frac{7}{4} + \frac{7}{4} \cdot 1 & \frac{27}{4} + \frac{7}{4} \cdot - 2 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.12.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 4 & - 12 \\ 0 & 1 & \frac{13}{8} & 0 & \frac{13}{4} \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.13

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

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Step 3.13.1

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

$[\begin{array}{c} 1 - 4 \cdot 0 & - 1 - 4 \cdot 0 & - 3 - 4 \cdot 0 & 4 - 4 \cdot 1 & - 12 - 4 \cdot - 2 \\ 0 & 1 & \frac{13}{8} & 0 & \frac{13}{4} \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.13.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 0 & - 4 \\ 0 & 1 & \frac{13}{8} & 0 & \frac{13}{4} \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.14

Perform the row operation $R_{2} = R_{2} - \frac{13}{8} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 3.14.1

Perform the row operation $R_{2} = R_{2} - \frac{13}{8} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 0 & - 4 \\ 0 - \frac{13}{8} \cdot 0 & 1 - \frac{13}{8} \cdot 0 & \frac{13}{8} - \frac{13}{8} \cdot 1 & 0 - \frac{13}{8} \cdot 0 & \frac{13}{4} - \frac{13}{8} \cdot 2 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & - 2 \end{array}]$

Step 3.14.2

Simplify $R_{2}$ .

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

Step 3.15

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

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Step 3.15.1

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

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

Step 3.15.2

Simplify $R_{1}$ .

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

Step 3.16

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

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Step 3.16.1

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

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

Step 3.16.2

Simplify $R_{1}$ .

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

Step 4

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

$w = 2$

$x = 0$

$y = 2$

$z = - 2$

Step 5

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

$(2, 0, 2, - 2)$