Finite Math Examples

$x - 2 y = 0$ , $x + y + z = 310$ , $45000 x + 40000 y + 80000 z = 16000000$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$

Step 2

Find the determinant of the coefficient matrix $[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}]$ .

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Step 2.1

Write $[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}]$ in determinant notation.

$| \begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array} |$

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.

$| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Step 2.2.4

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

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Step 2.2.5

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

$| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Step 2.2.6

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

$2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Step 2.2.7

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

$| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Step 2.2.8

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

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Step 2.2.9

Add the terms together.

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Step 2.3

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$ .

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Step 2.4

Evaluate $| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$ .

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Step 2.4.1

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

$1 (1 \cdot 80000 - 40000 \cdot 1) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

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 $80000$ by $1$ .

$1 (80000 - 40000 \cdot 1) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Step 2.4.2.1.2

Multiply $- 40000$ by $1$ .

$1 (80000 - 40000) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Step 2.4.2.2

Subtract $40000$ from $80000$ .

$1 \cdot 40000 + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Step 2.5

Evaluate $| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$ .

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Step 2.5.1

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

$1 \cdot 40000 + 2 (1 \cdot 80000 - 45000 \cdot 1) + 0$

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 $80000$ by $1$ .

$1 \cdot 40000 + 2 (80000 - 45000 \cdot 1) + 0$

Step 2.5.2.1.2

Multiply $- 45000$ by $1$ .

$1 \cdot 40000 + 2 (80000 - 45000) + 0$

Step 2.5.2.2

Subtract $45000$ from $80000$ .

$1 \cdot 40000 + 2 \cdot 35000 + 0$

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 $40000$ by $1$ .

$40000 + 2 \cdot 35000 + 0$

Step 2.6.1.2

Multiply $2$ by $35000$ .

$40000 + 70000 + 0$

Step 2.6.2

Add $40000$ and $70000$ .

$110000 + 0$

Step 2.6.3

Add $110000$ and $0$ .

$110000$

$D = 110000$

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 $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 0 & - 2 & 0 \\ 310 & 1 & 1 \\ 16000000 & 40000 & 80000 \end{array} |$

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.

$| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Step 4.2.1.4

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

$0 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Step 4.2.1.5

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

$| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Step 4.2.1.6

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

$2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Step 4.2.1.7

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

$| \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Step 4.2.1.8

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

$0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Step 4.2.1.9

Add the terms together.

$0 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Step 4.2.2

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$ .

$0 + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Step 4.2.3

Multiply $0$ by $| \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$ .

$0 + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0$

Step 4.2.4

Evaluate $| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$ .

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Step 4.2.4.1

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

$0 + 2 (310 \cdot 80000 - 16000000 \cdot 1) + 0$

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 $310$ by $80000$ .

$0 + 2 (24800000 - 16000000 \cdot 1) + 0$

Step 4.2.4.2.1.2

Multiply $- 16000000$ by $1$ .

$0 + 2 (24800000 - 16000000) + 0$

Step 4.2.4.2.2

Subtract $16000000$ from $24800000$ .

$0 + 2 \cdot 8800000 + 0$

Step 4.2.5

Simplify the determinant.

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Step 4.2.5.1

Multiply $2$ by $8800000$ .

$0 + 17600000 + 0$

Step 4.2.5.2

Add $0$ and $17600000$ .

$17600000 + 0$

Step 4.2.5.3

Add $17600000$ and $0$ .

$17600000$

$D_{x} = 17600000$

Step 4.3

Use the formula to solve for $x$ .

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

Step 4.4

Substitute $110000$ for $D$ and $17600000$ for $D_{x}$ in the formula.

$x = \frac{17600000}{110000}$

Step 4.5

Divide $17600000$ by $110000$ .

$x = 160$

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 $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 1 & 0 & 0 \\ 1 & 310 & 1 \\ 45000 & 16000000 & 80000 \end{array} |$

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.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Step 5.2.1.4

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

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Step 5.2.1.5

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

$| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Step 5.2.1.6

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

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Step 5.2.1.7

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

$| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Step 5.2.1.8

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

$0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Step 5.2.1.9

Add the terms together.

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Step 5.2.2

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$ .

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 + 0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Step 5.2.3

Multiply $0$ by $| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$ .

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 + 0$

Step 5.2.4

Evaluate $| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$ .

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Step 5.2.4.1

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

$1 (310 \cdot 80000 - 16000000 \cdot 1) + 0 + 0$

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 $310$ by $80000$ .

$1 (24800000 - 16000000 \cdot 1) + 0 + 0$

Step 5.2.4.2.1.2

Multiply $- 16000000$ by $1$ .

$1 (24800000 - 16000000) + 0 + 0$

Step 5.2.4.2.2

Subtract $16000000$ from $24800000$ .

$1 \cdot 8800000 + 0 + 0$

Step 5.2.5

Simplify the determinant.

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Step 5.2.5.1

Multiply $8800000$ by $1$ .

$8800000 + 0 + 0$

Step 5.2.5.2

Add $8800000$ and $0$ .

$8800000 + 0$

Step 5.2.5.3

Add $8800000$ and $0$ .

$8800000$

$D_{y} = 8800000$

Step 5.3

Use the formula to solve for $y$ .

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

Step 5.4

Substitute $110000$ for $D$ and $8800000$ for $D_{y}$ in the formula.

$y = \frac{8800000}{110000}$

Step 5.5

Divide $8800000$ by $110000$ .

$y = 80$

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 $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 310 \\ 45000 & 40000 & 16000000 \end{array} |$

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.

$| \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$

Step 6.2.1.4

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

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$

Step 6.2.1.5

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

$| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Step 6.2.1.6

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

$2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Step 6.2.1.7

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

$| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Step 6.2.1.8

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

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Step 6.2.1.9

Add the terms together.

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} | + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Step 6.2.2

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$ .

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} | + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Step 6.2.3

Evaluate $| \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$ .

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Step 6.2.3.1

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

$1 (1 \cdot 16000000 - 40000 \cdot 310) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Step 6.2.3.2

Simplify the determinant.

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Step 6.2.3.2.1

Simplify each term.

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Step 6.2.3.2.1.1

Multiply $16000000$ by $1$ .

$1 (16000000 - 40000 \cdot 310) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Step 6.2.3.2.1.2

Multiply $- 40000$ by $310$ .

$1 (16000000 - 12400000) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Step 6.2.3.2.2

Subtract $12400000$ from $16000000$ .

$1 \cdot 3600000 + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Step 6.2.4

Evaluate $| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$ .

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Step 6.2.4.1

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

$1 \cdot 3600000 + 2 (1 \cdot 16000000 - 45000 \cdot 310) + 0$

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 $16000000$ by $1$ .

$1 \cdot 3600000 + 2 (16000000 - 45000 \cdot 310) + 0$

Step 6.2.4.2.1.2

Multiply $- 45000$ by $310$ .

$1 \cdot 3600000 + 2 (16000000 - 13950000) + 0$

Step 6.2.4.2.2

Subtract $13950000$ from $16000000$ .

$1 \cdot 3600000 + 2 \cdot 2050000 + 0$

Step 6.2.5

Simplify the determinant.

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Step 6.2.5.1

Simplify each term.

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Step 6.2.5.1.1

Multiply $3600000$ by $1$ .

$3600000 + 2 \cdot 2050000 + 0$

Step 6.2.5.1.2

Multiply $2$ by $2050000$ .

$3600000 + 4100000 + 0$

Step 6.2.5.2

Add $3600000$ and $4100000$ .

$7700000 + 0$

Step 6.2.5.3

Add $7700000$ and $0$ .

$7700000$

$D_{z} = 7700000$

Step 6.3

Use the formula to solve for $z$ .

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

Step 6.4

Substitute $110000$ for $D$ and $7700000$ for $D_{z}$ in the formula.

$z = \frac{7700000}{110000}$

Step 6.5

Divide $7700000$ by $110000$ .

$z = 70$

Step 7

List the solution to the system of equations.

$x = 160$

$y = 80$

$z = 70$