Finite Math Examples

$y = 3 x + z - 2$ ,

$z = 3 x + 4$ ,

$y = 5 z$

Step 1

Move all of the variables to the left side of each equation.

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Step 1.1

Move all terms containing variables to the left side of the equation.

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Step 1.1.1

Subtract

$3 x$ from both sides of the equation.

$y - 3 x = z - 2$

$z = 3 x + 4$

$y = 5 z$

Step 1.1.2

Subtract

$z$ from both sides of the equation.

$y - 3 x - z = - 2$

$z = 3 x + 4$

$y = 5 z$

$y - 3 x - z = - 2$

$z = 3 x + 4$

$y = 5 z$

Step 1.2

Reorder

$y$ and

$- 3 x$ .

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

$z = 3 x + 4$

$y = 5 z$

Step 1.3

Subtract

$3 x$ from both sides of the equation.

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

$z - 3 x = 4$

$y = 5 z$

Step 1.4

Reorder

$z$ and

$- 3 x$ .

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

$- 3 x + z = 4$

$y = 5 z$

Step 1.5

Subtract

$5 z$ from both sides of the equation.

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

$- 3 x + z = 4$

$y - 5 z = 0$

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

$- 3 x + z = 4$

$y - 5 z = 0$

Step 2

Represent the system of equations in matrix format.

$[\begin{array}{c} - 3 & 1 & - 1 \\ - 3 & 0 & 1 \\ 0 & 1 & - 5 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 2 \\ 4 \\ 0 \end{array}]$

Step 3

Find the determinant of the coefficient matrix

$[\begin{array}{c} - 3 & 1 & - 1 \\ - 3 & 0 & 1 \\ 0 & 1 & - 5 \end{array}]$ .

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Step 3.1

Write

$[\begin{array}{c} - 3 & 1 & - 1 \\ - 3 & 0 & 1 \\ 0 & 1 & - 5 \end{array}]$ in determinant notation.

$| \begin{array}{c} - 3 & 1 & - 1 \\ - 3 & 0 & 1 \\ 0 & 1 & - 5 \end{array} |$

Step 3.2

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 column

$1$ by its cofactor and add.

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Step 3.2.1

Consider the corresponding sign chart.

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

Step 3.2.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 3.2.3

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

$| \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} |$

Step 3.2.4

Multiply element

$a_{11}$ by its cofactor.

$- 3 | \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} |$

Step 3.2.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} |$

Step 3.2.6

Multiply element

$a_{21}$ by its cofactor.

$3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} |$

Step 3.2.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

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

Step 3.2.8

Multiply element

$a_{31}$ by its cofactor.

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

Step 3.2.9

Add the terms together.

$- 3 | \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} | + 3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0 | \begin{array}{c} 1 & - 1 \\ 0 & 1 \end{array} |$

Step 3.3

Multiply

$0$ by

$| \begin{array}{c} 1 & - 1 \\ 0 & 1 \end{array} |$ .

$- 3 | \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} | + 3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 3.4

Evaluate

$| \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} |$ .

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Step 3.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3 (0 \cdot - 5 - 1 \cdot 1) + 3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 3.4.2

Simplify the determinant.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply

$0$ by

$- 5$ .

$- 3 (0 - 1 \cdot 1) + 3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 3.4.2.1.2

Multiply

$- 1$ by

$1$ .

$- 3 (0 - 1) + 3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 3.4.2.2

Subtract

$1$ from

$0$ .

$- 3 \cdot - 1 + 3 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 3.5

Evaluate

$| \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} |$ .

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Step 3.5.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3 \cdot - 1 + 3 (1 \cdot - 5 - 1 \cdot - 1) + 0$

Step 3.5.2

Simplify the determinant.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Multiply

$- 5$ by

$1$ .

$- 3 \cdot - 1 + 3 (- 5 - 1 \cdot - 1) + 0$

Step 3.5.2.1.2

Multiply

$- 1$ by

$- 1$ .

$- 3 \cdot - 1 + 3 (- 5 + 1) + 0$

Step 3.5.2.2

Add

$- 5$ and

$1$ .

$- 3 \cdot - 1 + 3 \cdot - 4 + 0$

Step 3.6

Simplify the determinant.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

Multiply

$- 3$ by

$- 1$ .

$3 + 3 \cdot - 4 + 0$

Step 3.6.1.2

Multiply

$3$ by

$- 4$ .

$3 - 12 + 0$

Step 3.6.2

Subtract

$12$ from

$3$ .

$- 9 + 0$

Step 3.6.3

Add

$- 9$ and

$0$ .

$- 9$

$D = - 9$

Step 4

Since the determinant is not

$0$ , the system can be solved using Cramer's Rule.

Step 5

Find the value of

$x$ by Cramer's Rule, which states that

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

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Step 5.1

Replace column

$1$ of the coefficient matrix that corresponds to the

$x$ -coefficients of the system with

$[\begin{array}{c} - 2 \\ 4 \\ 0 \end{array}]$ .

$| \begin{array}{c} - 2 & 1 & - 1 \\ 4 & 0 & 1 \\ 0 & 1 & - 5 \end{array} |$

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 column

$1$ 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.

$| \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} |$

Step 5.2.1.4

Multiply element

$a_{11}$ by its cofactor.

$- 2 | \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} |$

Step 5.2.1.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} |$

Step 5.2.1.6

Multiply element

$a_{21}$ by its cofactor.

$- 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} |$

Step 5.2.1.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

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

Step 5.2.1.8

Multiply element

$a_{31}$ by its cofactor.

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

Step 5.2.1.9

Add the terms together.

$- 2 | \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} | - 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0 | \begin{array}{c} 1 & - 1 \\ 0 & 1 \end{array} |$

Step 5.2.2

Multiply

$0$ by

$| \begin{array}{c} 1 & - 1 \\ 0 & 1 \end{array} |$ .

$- 2 | \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} | - 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 5.2.3

Evaluate

$| \begin{array}{c} 0 & 1 \\ 1 & - 5 \end{array} |$ .

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Step 5.2.3.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 2 (0 \cdot - 5 - 1 \cdot 1) - 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

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

$0$ by

$- 5$ .

$- 2 (0 - 1 \cdot 1) - 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 5.2.3.2.1.2

Multiply

$- 1$ by

$1$ .

$- 2 (0 - 1) - 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 5.2.3.2.2

Subtract

$1$ from

$0$ .

$- 2 \cdot - 1 - 4 | \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} | + 0$

Step 5.2.4

Evaluate

$| \begin{array}{c} 1 & - 1 \\ 1 & - 5 \end{array} |$ .

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Step 5.2.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 2 \cdot - 1 - 4 (1 \cdot - 5 - 1 \cdot - 1) + 0$

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

$- 5$ by

$1$ .

$- 2 \cdot - 1 - 4 (- 5 - 1 \cdot - 1) + 0$

Step 5.2.4.2.1.2

Multiply

$- 1$ by

$- 1$ .

$- 2 \cdot - 1 - 4 (- 5 + 1) + 0$

Step 5.2.4.2.2

Add

$- 5$ and

$1$ .

$- 2 \cdot - 1 - 4 \cdot - 4 + 0$

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

$- 2$ by

$- 1$ .

$2 - 4 \cdot - 4 + 0$

Step 5.2.5.1.2

Multiply

$- 4$ by

$- 4$ .

$2 + 16 + 0$

Step 5.2.5.2

Add

$2$ and

$16$ .

$18 + 0$

Step 5.2.5.3

Add

$18$ and

$0$ .

$18$

$D_{x} = 18$

Step 5.3

Use the formula to solve for

$x$ .

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

Step 5.4

Substitute

$- 9$ for

$D$ and

$18$ for

$D_{x}$ in the formula.

$x = \frac{18}{- 9}$

Step 5.5

Divide

$18$ by

$- 9$ .

$x = - 2$

Step 6

Find the value of

$y$ by Cramer's Rule, which states that

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

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Step 6.1

Replace column

$2$ of the coefficient matrix that corresponds to the

$y$ -coefficients of the system with

$[\begin{array}{c} - 2 \\ 4 \\ 0 \end{array}]$ .

$| \begin{array}{c} - 3 & - 2 & - 1 \\ - 3 & 4 & 1 \\ 0 & 0 & - 5 \end{array} |$

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

$3$ 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

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

$| \begin{array}{c} - 2 & - 1 \\ 4 & 1 \end{array} |$

Step 6.2.1.4

Multiply element

$a_{31}$ by its cofactor.

$0 | \begin{array}{c} - 2 & - 1 \\ 4 & 1 \end{array} |$

Step 6.2.1.5

The minor for

$a_{32}$ is the determinant with row

$3$ and column

$2$ deleted.

$| \begin{array}{c} - 3 & - 1 \\ - 3 & 1 \end{array} |$

Step 6.2.1.6

Multiply element

$a_{32}$ by its cofactor.

$0 | \begin{array}{c} - 3 & - 1 \\ - 3 & 1 \end{array} |$

Step 6.2.1.7

The minor for

$a_{33}$ is the determinant with row

$3$ and column

$3$ deleted.

$| \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 6.2.1.8

Multiply element

$a_{33}$ by its cofactor.

$- 5 | \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 6.2.1.9

Add the terms together.

$0 | \begin{array}{c} - 2 & - 1 \\ 4 & 1 \end{array} | + 0 | \begin{array}{c} - 3 & - 1 \\ - 3 & 1 \end{array} | - 5 | \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 6.2.2

Multiply

$0$ by

$| \begin{array}{c} - 2 & - 1 \\ 4 & 1 \end{array} |$ .

$0 + 0 | \begin{array}{c} - 3 & - 1 \\ - 3 & 1 \end{array} | - 5 | \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 6.2.3

Multiply

$0$ by

$| \begin{array}{c} - 3 & - 1 \\ - 3 & 1 \end{array} |$ .

$0 + 0 - 5 | \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 6.2.4

Evaluate

$| \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$ .

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Step 6.2.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 + 0 - 5 (- 3 \cdot 4 - (- 3 \cdot - 2))$

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

$- 3$ by

$4$ .

$0 + 0 - 5 (- 12 - (- 3 \cdot - 2))$

Step 6.2.4.2.1.2

Multiply

$- (- 3 \cdot - 2)$ .

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Step 6.2.4.2.1.2.1

Multiply

$- 3$ by

$- 2$ .

$0 + 0 - 5 (- 12 - 1 \cdot 6)$

Step 6.2.4.2.1.2.2

Multiply

$- 1$ by

$6$ .

$0 + 0 - 5 (- 12 - 6)$

Step 6.2.4.2.2

Subtract

$6$ from

$- 12$ .

$0 + 0 - 5 \cdot - 18$

Step 6.2.5

Simplify the determinant.

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Step 6.2.5.1

Multiply

$- 5$ by

$- 18$ .

$0 + 0 + 90$

Step 6.2.5.2

Add

$0$ and

$0$ .

$0 + 90$

Step 6.2.5.3

Add

$0$ and

$90$ .

$90$

$D_{y} = 90$

Step 6.3

Use the formula to solve for

$y$ .

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

Step 6.4

Substitute

$- 9$ for

$D$ and

$90$ for

$D_{y}$ in the formula.

$y = \frac{90}{- 9}$

Step 6.5

Divide

$90$ by

$- 9$ .

$y = - 10$

Step 7

Find the value of

$z$ by Cramer's Rule, which states that

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

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Step 7.1

Replace column

$3$ of the coefficient matrix that corresponds to the

$z$ -coefficients of the system with

$[\begin{array}{c} - 2 \\ 4 \\ 0 \end{array}]$ .

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

Step 7.2

Find the determinant.

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Step 7.2.1

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

$3$ by its cofactor and add.

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Step 7.2.1.1

Consider the corresponding sign chart.

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

Step 7.2.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 7.2.1.3

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

$| \begin{array}{c} 1 & - 2 \\ 0 & 4 \end{array} |$

Step 7.2.1.4

Multiply element

$a_{31}$ by its cofactor.

$0 | \begin{array}{c} 1 & - 2 \\ 0 & 4 \end{array} |$

Step 7.2.1.5

The minor for

$a_{32}$ is the determinant with row

$3$ and column

$2$ deleted.

$| \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 7.2.1.6

Multiply element

$a_{32}$ by its cofactor.

$- 1 | \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$

Step 7.2.1.7

The minor for

$a_{33}$ is the determinant with row

$3$ and column

$3$ deleted.

$| \begin{array}{c} - 3 & 1 \\ - 3 & 0 \end{array} |$

Step 7.2.1.8

Multiply element

$a_{33}$ by its cofactor.

$0 | \begin{array}{c} - 3 & 1 \\ - 3 & 0 \end{array} |$

Step 7.2.1.9

Add the terms together.

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

Step 7.2.2

Multiply

$0$ by

$| \begin{array}{c} 1 & - 2 \\ 0 & 4 \end{array} |$ .

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

Step 7.2.3

Multiply

$0$ by

$| \begin{array}{c} - 3 & 1 \\ - 3 & 0 \end{array} |$ .

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

Step 7.2.4

Evaluate

$| \begin{array}{c} - 3 & - 2 \\ - 3 & 4 \end{array} |$ .

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Step 7.2.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 - 1 (- 3 \cdot 4 - (- 3 \cdot - 2)) + 0$

Step 7.2.4.2

Simplify the determinant.

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Step 7.2.4.2.1

Simplify each term.

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Step 7.2.4.2.1.1

Multiply

$- 3$ by

$4$ .

$0 - 1 (- 12 - (- 3 \cdot - 2)) + 0$

Step 7.2.4.2.1.2

Multiply

$- (- 3 \cdot - 2)$ .

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Step 7.2.4.2.1.2.1

Multiply

$- 3$ by

$- 2$ .

$0 - 1 (- 12 - 1 \cdot 6) + 0$

Step 7.2.4.2.1.2.2

Multiply

$- 1$ by

$6$ .

$0 - 1 (- 12 - 6) + 0$

Step 7.2.4.2.2

Subtract

$6$ from

$- 12$ .

$0 - 1 \cdot - 18 + 0$

Step 7.2.5

Simplify the determinant.

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Step 7.2.5.1

Multiply

$- 1$ by

$- 18$ .

$0 + 18 + 0$

Step 7.2.5.2

Add

$0$ and

$18$ .

$18 + 0$

Step 7.2.5.3

Add

$18$ and

$0$ .

$18$

$D_{z} = 18$

Step 7.3

Use the formula to solve for

$z$ .

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

Step 7.4

Substitute

$- 9$ for

$D$ and

$18$ for

$D_{z}$ in the formula.

$z = \frac{18}{- 9}$

Step 7.5

Divide

$18$ by

$- 9$ .

$z = - 2$

Step 8

List the solution to the system of equations.

$x = - 2$

$y = - 10$

$z = - 2$

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