Finite Math Examples

$x - 3 y + 3 z = 45$ , $y - z + w = - 11$ , $- 2 x + 2 y - 3 z + 4 w = 28$ , $3 y - 3 z + w = - 47$

Step 1

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

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Step 1.1

Move $- z$ .

$x - 3 y + 3 z = 45$

$y + w - z = - 11$

$- 2 x + 2 y - 3 z + 4 w = 28$

$3 y - 3 z + w = - 47$

Step 1.2

Reorder $y$ and $w$ .

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$- 2 x + 2 y - 3 z + 4 w = 28$

$3 y - 3 z + w = - 47$

Step 1.3

Move $- 3 z$ .

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$- 2 x + 2 y + 4 w - 3 z = 28$

$3 y - 3 z + w = - 47$

Step 1.4

Move $2 y$ .

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$- 2 x + 4 w + 2 y - 3 z = 28$

$3 y - 3 z + w = - 47$

Step 1.5

Reorder $- 2 x$ and $4 w$ .

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$4 w - 2 x + 2 y - 3 z = 28$

$3 y - 3 z + w = - 47$

Step 1.6

Move $- 3 z$ .

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$4 w - 2 x + 2 y - 3 z = 28$

$3 y + w - 3 z = - 47$

Step 1.7

Reorder $3 y$ and $w$ .

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$4 w - 2 x + 2 y - 3 z = 28$

$w + 3 y - 3 z = - 47$

$x - 3 y + 3 z = 45$

$w + y - z = - 11$

$4 w - 2 x + 2 y - 3 z = 28$

$w + 3 y - 3 z = - 47$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 0 & 1 & - 3 & 3 & 45 \\ 1 & 0 & 1 & - 1 & - 11 \\ 4 & - 2 & 2 & - 3 & 28 \\ 1 & 0 & 3 & - 3 & - 47 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 4 & - 2 & 2 & - 3 & 28 \\ 1 & 0 & 3 & - 3 & - 47 \end{array}]$

Step 3.2

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.2.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 4 - 4 \cdot 1 & - 2 - 4 \cdot 0 & 2 - 4 \cdot 1 & - 3 - 4 \cdot - 1 & 28 - 4 \cdot - 11 \\ 1 & 0 & 3 & - 3 & - 47 \end{array}]$

Step 3.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & - 2 & - 2 & 1 & 72 \\ 1 & 0 & 3 & - 3 & - 47 \end{array}]$

Step 3.3

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

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Step 3.3.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & - 2 & - 2 & 1 & 72 \\ 1 - 1 & 0 - 0 & 3 - 1 & - 3 + 1 & - 47 + 11 \end{array}]$

Step 3.3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & - 2 & - 2 & 1 & 72 \\ 0 & 0 & 2 & - 2 & - 36 \end{array}]$

Step 3.4

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

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Step 3.4.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 2 + 2 \cdot - 3 & 1 + 2 \cdot 3 & 72 + 2 \cdot 45 \\ 0 & 0 & 2 & - 2 & - 36 \end{array}]$

Step 3.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & - 8 & 7 & 162 \\ 0 & 0 & 2 & - 2 & - 36 \end{array}]$

Step 3.5

Multiply each element of $R_{3}$ by $- \frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

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Step 3.5.1

Multiply each element of $R_{3}$ by $- \frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 & - \frac{1}{8} \cdot 7 & - \frac{1}{8} \cdot 162 \\ 0 & 0 & 2 & - 2 & - 36 \end{array}]$

Step 3.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & 1 & - \frac{7}{8} & - \frac{81}{4} \\ 0 & 0 & 2 & - 2 & - 36 \end{array}]$

Step 3.6

Perform the row operation $R_{4} = R_{4} - 2 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 3.6.1

Perform the row operation $R_{4} = R_{4} - 2 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & 1 & - \frac{7}{8} & - \frac{81}{4} \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 2 - 2 (- \frac{7}{8}) & - 36 - 2 (- \frac{81}{4}) \end{array}]$

Step 3.6.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & 1 & - \frac{7}{8} & - \frac{81}{4} \\ 0 & 0 & 0 & - \frac{1}{4} & \frac{9}{2} \end{array}]$

Step 3.7

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

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Step 3.7.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & 1 & - \frac{7}{8} & - \frac{81}{4} \\ - 4 \cdot 0 & - 4 \cdot 0 & - 4 \cdot 0 & - 4 (- \frac{1}{4}) & - 4 (\frac{9}{2}) \end{array}]$

Step 3.7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & 1 & - \frac{7}{8} & - \frac{81}{4} \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.8

Perform the row operation $R_{3} = R_{3} + \frac{7}{8} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 3.8.1

Perform the row operation $R_{3} = R_{3} + \frac{7}{8} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 + \frac{7}{8} \cdot 0 & 0 + \frac{7}{8} \cdot 0 & 1 + \frac{7}{8} \cdot 0 & - \frac{7}{8} + \frac{7}{8} \cdot 1 & - \frac{81}{4} + \frac{7}{8} \cdot - 18 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 3 & 45 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.9

Perform the row operation $R_{2} = R_{2} - 3 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 3.9.1

Perform the row operation $R_{2} = R_{2} - 3 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & - 3 - 3 \cdot 0 & 3 - 3 \cdot 1 & 45 - 3 \cdot - 18 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.9.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 11 \\ 0 & 1 & - 3 & 0 & 99 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.10

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

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Step 3.10.1

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

$[\begin{array}{c} 1 + 0 & 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - 11 - 18 \\ 0 & 1 & - 3 & 0 & 99 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.10.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 29 \\ 0 & 1 & - 3 & 0 & 99 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.11

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

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Step 3.11.1

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

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 29 \\ 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 & 99 + 3 \cdot - 36 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.11.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 29 \\ 0 & 1 & 0 & 0 & - 9 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.12

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

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Step 3.12.1

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

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & 0 - 0 & - 29 + 36 \\ 0 & 1 & 0 & 0 & - 9 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 3.12.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 7 \\ 0 & 1 & 0 & 0 & - 9 \\ 0 & 0 & 1 & 0 & - 36 \\ 0 & 0 & 0 & 1 & - 18 \end{array}]$

Step 4

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

$w = 7$

$x = - 9$

$y = - 36$

$z = - 18$

Step 5

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

$(7, - 9, - 36, - 18)$