Finite Math Examples

$x - z = 0$ , $x + y + z = 94$ , $77 x + 54 y + 66 z = 5811$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} 1 & 0 & - 1 \\ 1 & 1 & 1 \\ 77 & 54 & 66 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 94 \\ 5811 \end{array}]$

Step 2

Find the determinant of the coefficient matrix $[\begin{array}{c} 1 & 0 & - 1 \\ 1 & 1 & 1 \\ 77 & 54 & 66 \end{array}]$ .

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Step 2.1

Write $[\begin{array}{c} 1 & 0 & - 1 \\ 1 & 1 & 1 \\ 77 & 54 & 66 \end{array}]$ in determinant notation.

$| \begin{array}{c} 1 & 0 & - 1 \\ 1 & 1 & 1 \\ 77 & 54 & 66 \end{array} |$

Step 2.2

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.2.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.2.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.2.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} |$

Step 2.2.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} |$

Step 2.2.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} |$

Step 2.2.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} |$

Step 2.2.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.2.8

Multiply element $a_{13}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.2.9

Add the terms together.

$1 | \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} | - 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.3

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} |$ .

$1 | \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} | + 0 - 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.4

Evaluate $| \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} |$ .

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Step 2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (1 \cdot 66 - 54 \cdot 1) + 0 - 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.4.2

Simplify the determinant.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $66$ by $1$ .

$1 (66 - 54 \cdot 1) + 0 - 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.4.2.1.2

Multiply $- 54$ by $1$ .

$1 (66 - 54) + 0 - 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.4.2.2

Subtract $54$ from $66$ .

$1 \cdot 12 + 0 - 1 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 2.5

Evaluate $| \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$ .

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Step 2.5.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot 12 + 0 - 1 (1 \cdot 54 - 77 \cdot 1)$

Step 2.5.2

Simplify the determinant.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $54$ by $1$ .

$1 \cdot 12 + 0 - 1 (54 - 77 \cdot 1)$

Step 2.5.2.1.2

Multiply $- 77$ by $1$ .

$1 \cdot 12 + 0 - 1 (54 - 77)$

Step 2.5.2.2

Subtract $77$ from $54$ .

$1 \cdot 12 + 0 - 1 \cdot - 23$

Step 2.6

Simplify the determinant.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Multiply $12$ by $1$ .

$12 + 0 - 1 \cdot - 23$

Step 2.6.1.2

Multiply $- 1$ by $- 23$ .

$12 + 0 + 23$

Step 2.6.2

Add $12$ and $0$ .

$12 + 23$

Step 2.6.3

Add $12$ and $23$ .

$35$

$D = 35$

Step 3

Since the determinant is not $0$ , the system can be solved using Cramer's Rule.

Step 4

Find the value of $x$ by Cramer's Rule, which states that $x = \frac{D_{x}}{D}$ .

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Step 4.1

Replace column $1$ of the coefficient matrix that corresponds to the $x$ -coefficients of the system with $[\begin{array}{c} 0 \\ 94 \\ 5811 \end{array}]$ .

$| \begin{array}{c} 0 & 0 & - 1 \\ 94 & 1 & 1 \\ 5811 & 54 & 66 \end{array} |$

Step 4.2

Find the determinant.

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Step 4.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 4.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 4.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 4.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} |$

Step 4.2.1.4

Multiply element $a_{11}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} |$

Step 4.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} |$

Step 4.2.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} |$

Step 4.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 94 & 1 \\ 5811 & 54 \end{array} |$

Step 4.2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 1 | \begin{array}{c} 94 & 1 \\ 5811 & 54 \end{array} |$

Step 4.2.1.9

Add the terms together.

$0 | \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} | + 0 | \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} | - 1 | \begin{array}{c} 94 & 1 \\ 5811 & 54 \end{array} |$

Step 4.2.2

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 54 & 66 \end{array} |$ .

$0 + 0 | \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} | - 1 | \begin{array}{c} 94 & 1 \\ 5811 & 54 \end{array} |$

Step 4.2.3

Multiply $0$ by $| \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} |$ .

$0 + 0 - 1 | \begin{array}{c} 94 & 1 \\ 5811 & 54 \end{array} |$

Step 4.2.4

Evaluate $| \begin{array}{c} 94 & 1 \\ 5811 & 54 \end{array} |$ .

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Step 4.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 + 0 - 1 (94 \cdot 54 - 5811 \cdot 1)$

Step 4.2.4.2

Simplify the determinant.

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Step 4.2.4.2.1

Simplify each term.

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Step 4.2.4.2.1.1

Multiply $94$ by $54$ .

$0 + 0 - 1 (5076 - 5811 \cdot 1)$

Step 4.2.4.2.1.2

Multiply $- 5811$ by $1$ .

$0 + 0 - 1 (5076 - 5811)$

Step 4.2.4.2.2

Subtract $5811$ from $5076$ .

$0 + 0 - 1 \cdot - 735$

Step 4.2.5

Simplify the determinant.

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Step 4.2.5.1

Multiply $- 1$ by $- 735$ .

$0 + 0 + 735$

Step 4.2.5.2

Add $0$ and $0$ .

$0 + 735$

Step 4.2.5.3

Add $0$ and $735$ .

$735$

$D_{x} = 735$

Step 4.3

Use the formula to solve for $x$ .

$x = \frac{D_{x}}{D}$

Step 4.4

Substitute $35$ for $D$ and $735$ for $D_{x}$ in the formula.

$x = \frac{735}{35}$

Step 4.5

Divide $735$ by $35$ .

$x = 21$

Step 5

Find the value of $y$ by Cramer's Rule, which states that $y = \frac{D_{y}}{D}$ .

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Step 5.1

Replace column $2$ of the coefficient matrix that corresponds to the $y$ -coefficients of the system with $[\begin{array}{c} 0 \\ 94 \\ 5811 \end{array}]$ .

$| \begin{array}{c} 1 & 0 & - 1 \\ 1 & 94 & 1 \\ 77 & 5811 & 66 \end{array} |$

Step 5.2

Find the determinant.

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Step 5.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 5.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} |$

Step 5.2.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} |$

Step 5.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} |$

Step 5.2.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} |$

Step 5.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.1.9

Add the terms together.

$1 | \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} | - 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.2

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 77 & 66 \end{array} |$ .

$1 | \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} | + 0 - 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.3

Evaluate $| \begin{array}{c} 94 & 1 \\ 5811 & 66 \end{array} |$ .

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Step 5.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (94 \cdot 66 - 5811 \cdot 1) + 0 - 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.3.2

Simplify the determinant.

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Step 5.2.3.2.1

Simplify each term.

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Step 5.2.3.2.1.1

Multiply $94$ by $66$ .

$1 (6204 - 5811 \cdot 1) + 0 - 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.3.2.1.2

Multiply $- 5811$ by $1$ .

$1 (6204 - 5811) + 0 - 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.3.2.2

Subtract $5811$ from $6204$ .

$1 \cdot 393 + 0 - 1 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 5.2.4

Evaluate $| \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$ .

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Step 5.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot 393 + 0 - 1 (1 \cdot 5811 - 77 \cdot 94)$

Step 5.2.4.2

Simplify the determinant.

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Step 5.2.4.2.1

Simplify each term.

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Step 5.2.4.2.1.1

Multiply $5811$ by $1$ .

$1 \cdot 393 + 0 - 1 (5811 - 77 \cdot 94)$

Step 5.2.4.2.1.2

Multiply $- 77$ by $94$ .

$1 \cdot 393 + 0 - 1 (5811 - 7238)$

Step 5.2.4.2.2

Subtract $7238$ from $5811$ .

$1 \cdot 393 + 0 - 1 \cdot - 1427$

Step 5.2.5

Simplify the determinant.

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Step 5.2.5.1

Simplify each term.

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Step 5.2.5.1.1

Multiply $393$ by $1$ .

$393 + 0 - 1 \cdot - 1427$

Step 5.2.5.1.2

Multiply $- 1$ by $- 1427$ .

$393 + 0 + 1427$

Step 5.2.5.2

Add $393$ and $0$ .

$393 + 1427$

Step 5.2.5.3

Add $393$ and $1427$ .

$1820$

$D_{y} = 1820$

Step 5.3

Use the formula to solve for $y$ .

$y = \frac{D_{y}}{D}$

Step 5.4

Substitute $35$ for $D$ and $1820$ for $D_{y}$ in the formula.

$y = \frac{1820}{35}$

Step 5.5

Divide $1820$ by $35$ .

$y = 52$

Step 6

Find the value of $z$ by Cramer's Rule, which states that $z = \frac{D_{z}}{D}$ .

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Step 6.1

Replace column $3$ of the coefficient matrix that corresponds to the $z$ -coefficients of the system with $[\begin{array}{c} 0 \\ 94 \\ 5811 \end{array}]$ .

$| \begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 94 \\ 77 & 54 & 5811 \end{array} |$

Step 6.2

Find the determinant.

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Step 6.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 6.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 6.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 6.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 94 \\ 54 & 5811 \end{array} |$

Step 6.2.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 94 \\ 54 & 5811 \end{array} |$

Step 6.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 6.2.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$

Step 6.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 6.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 6.2.1.9

Add the terms together.

$1 | \begin{array}{c} 1 & 94 \\ 54 & 5811 \end{array} | + 0 | \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 6.2.2

Multiply $0$ by $| \begin{array}{c} 1 & 94 \\ 77 & 5811 \end{array} |$ .

$1 | \begin{array}{c} 1 & 94 \\ 54 & 5811 \end{array} | + 0 + 0 | \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$

Step 6.2.3

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 77 & 54 \end{array} |$ .

$1 | \begin{array}{c} 1 & 94 \\ 54 & 5811 \end{array} | + 0 + 0$

Step 6.2.4

Evaluate $| \begin{array}{c} 1 & 94 \\ 54 & 5811 \end{array} |$ .

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Step 6.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (1 \cdot 5811 - 54 \cdot 94) + 0 + 0$

Step 6.2.4.2

Simplify the determinant.

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Step 6.2.4.2.1

Simplify each term.

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Step 6.2.4.2.1.1

Multiply $5811$ by $1$ .

$1 (5811 - 54 \cdot 94) + 0 + 0$

Step 6.2.4.2.1.2

Multiply $- 54$ by $94$ .

$1 (5811 - 5076) + 0 + 0$

Step 6.2.4.2.2

Subtract $5076$ from $5811$ .

$1 \cdot 735 + 0 + 0$

Step 6.2.5

Simplify the determinant.

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Step 6.2.5.1

Multiply $735$ by $1$ .

$735 + 0 + 0$

Step 6.2.5.2

Add $735$ and $0$ .

$735 + 0$

Step 6.2.5.3

Add $735$ and $0$ .

$735$

$D_{z} = 735$

Step 6.3

Use the formula to solve for $z$ .

$z = \frac{D_{z}}{D}$

Step 6.4

Substitute $35$ for $D$ and $735$ for $D_{z}$ in the formula.

$z = \frac{735}{35}$

Step 6.5

Divide $735$ by $35$ .

$z = 21$

Step 7

List the solution to the system of equations.

$x = 21$

$y = 52$

$z = 21$