Finite Math Examples

$x + 2 y + z = 60$ , $2 x + y + z = 48$ , $3 x + y + 2 z = 42$ , $72 x + 66 y + 102 z = 0$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 2 & 1 & 1 & 48 \\ 3 & 1 & 2 & 42 \\ 72 & 66 & 102 & 0 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 2 & 1 - 2 \cdot 1 & 48 - 2 \cdot 60 \\ 3 & 1 & 2 & 42 \\ 72 & 66 & 102 & 0 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & - 3 & - 1 & - 72 \\ 3 & 1 & 2 & 42 \\ 72 & 66 & 102 & 0 \end{array}]$

Step 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & - 3 & - 1 & - 72 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot 2 & 2 - 3 \cdot 1 & 42 - 3 \cdot 60 \\ 72 & 66 & 102 & 0 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & - 3 & - 1 & - 72 \\ 0 & - 5 & - 1 & - 138 \\ 72 & 66 & 102 & 0 \end{array}]$

Step 2.3

Perform the row operation $R_{4} = R_{4} - 72 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 2.3.1

Perform the row operation $R_{4} = R_{4} - 72 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & - 3 & - 1 & - 72 \\ 0 & - 5 & - 1 & - 138 \\ 72 - 72 \cdot 1 & 66 - 72 \cdot 2 & 102 - 72 \cdot 1 & 0 - 72 \cdot 60 \end{array}]$

Step 2.3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & - 3 & - 1 & - 72 \\ 0 & - 5 & - 1 & - 138 \\ 0 & - 78 & 30 & - 4320 \end{array}]$

Step 2.4

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

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Step 2.4.1

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

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 1 & - \frac{1}{3} \cdot - 72 \\ 0 & - 5 & - 1 & - 138 \\ 0 & - 78 & 30 & - 4320 \end{array}]$

Step 2.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & - 5 & - 1 & - 138 \\ 0 & - 78 & 30 & - 4320 \end{array}]$

Step 2.5

Perform the row operation $R_{3} = R_{3} + 5 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 2.5.1

Perform the row operation $R_{3} = R_{3} + 5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 1 + 5 (\frac{1}{3}) & - 138 + 5 \cdot 24 \\ 0 & - 78 & 30 & - 4320 \end{array}]$

Step 2.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & \frac{2}{3} & - 18 \\ 0 & - 78 & 30 & - 4320 \end{array}]$

Step 2.6

Perform the row operation $R_{4} = R_{4} + 78 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 2.6.1

Perform the row operation $R_{4} = R_{4} + 78 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & \frac{2}{3} & - 18 \\ 0 + 78 \cdot 0 & - 78 + 78 \cdot 1 & 30 + 78 (\frac{1}{3}) & - 4320 + 78 \cdot 24 \end{array}]$

Step 2.6.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & \frac{2}{3} & - 18 \\ 0 & 0 & 56 & - 2448 \end{array}]$

Step 2.7

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

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Step 2.7.1

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

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot \frac{2}{3} & \frac{3}{2} \cdot - 18 \\ 0 & 0 & 56 & - 2448 \end{array}]$

Step 2.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & 1 & - 27 \\ 0 & 0 & 56 & - 2448 \end{array}]$

Step 2.8

Perform the row operation $R_{4} = R_{4} - 56 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 2.8.1

Perform the row operation $R_{4} = R_{4} - 56 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & 1 & - 27 \\ 0 - 56 \cdot 0 & 0 - 56 \cdot 0 & 56 - 56 \cdot 1 & - 2448 - 56 \cdot - 27 \end{array}]$

Step 2.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & 1 & - 27 \\ 0 & 0 & 0 & - 936 \end{array}]$

Step 2.9

Multiply each element of $R_{4}$ by $- \frac{1}{936}$ to make the entry at $4, 4$ a $1$ .

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Step 2.9.1

Multiply each element of $R_{4}$ by $- \frac{1}{936}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & 1 & - 27 \\ - \frac{1}{936} \cdot 0 & - \frac{1}{936} \cdot 0 & - \frac{1}{936} \cdot 0 & - \frac{1}{936} \cdot - 936 \end{array}]$

Step 2.9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & 1 & - 27 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 2.10

Perform the row operation $R_{3} = R_{3} + 27 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 2.10.1

Perform the row operation $R_{3} = R_{3} + 27 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 + 27 \cdot 0 & 0 + 27 \cdot 0 & 1 + 27 \cdot 0 & - 27 + 27 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 2.10.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 24 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 2.11

Perform the row operation $R_{2} = R_{2} - 24 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 2.11.1

Perform the row operation $R_{2} = R_{2} - 24 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 - 24 \cdot 0 & 1 - 24 \cdot 0 & \frac{1}{3} - 24 \cdot 0 & 24 - 24 \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 2.11.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 1 & 60 \\ 0 & 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 2.12

Perform the row operation $R_{1} = R_{1} - 60 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Step 2.12.1

Perform the row operation $R_{1} = R_{1} - 60 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 60 \cdot 0 & 2 - 60 \cdot 0 & 1 - 60 \cdot 0 & 60 - 60 \cdot 1 \\ 0 & 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 2.12.2

Simplify $R_{1}$ .

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

Step 2.13

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

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Step 2.13.1

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

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

Step 2.13.2

Simplify $R_{2}$ .

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

Step 2.14

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

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Step 2.14.1

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

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

Step 2.14.2

Simplify $R_{1}$ .

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

Step 2.15

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

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Step 2.15.1

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

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

Step 2.15.2

Simplify $R_{1}$ .

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

Step 3

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

$x = 0$

$y = 0$

$z = 0$

$0 = 1$

Step 4

The system is inconsistent so there is no solution.

$No solution$