Precalculus Examples

$\frac{7}{4} x - \frac{9}{4} y - z = - 6$ , $x - y - z = - 5$ , $- \frac{5}{4} x + \frac{7}{4} y + z = 1$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} \frac{7}{4} & - \frac{9}{4} & - 1 \\ 1 & - 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 6 \\ - 5 \\ 1 \end{array}]$

Step 2

Find the determinant of the coefficient matrix $[\begin{array}{c} \frac{7}{4} & - \frac{9}{4} & - 1 \\ 1 & - 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} & 1 \end{array}]$ .

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Step 2.1

Write $[\begin{array}{c} \frac{7}{4} & - \frac{9}{4} & - 1 \\ 1 & - 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} & 1 \end{array}]$ in determinant notation.

$| \begin{array}{c} \frac{7}{4} & - \frac{9}{4} & - 1 \\ 1 & - 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} & 1 \end{array} |$

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.

$| \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} |$

Step 2.2.4

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

$\frac{7}{4} | \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} |$

Step 2.2.5

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

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

Step 2.2.6

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

$\frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} |$

Step 2.2.7

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

$| \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.2.8

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

$- 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.2.9

Add the terms together.

$\frac{7}{4} | \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} | + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3

Evaluate $| \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} |$ .

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Step 2.3.1

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

$\frac{7}{4} (- 1 \cdot 1 - \frac{7}{4} \cdot - 1) + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2

Simplify the determinant.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $- 1$ by $1$ .

$\frac{7}{4} (- 1 - \frac{7}{4} \cdot - 1) + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.1.2

Multiply $- \frac{7}{4} \cdot - 1$ .

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Step 2.3.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{7}{4} (- 1 + 1 (\frac{7}{4})) + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.1.2.2

Multiply $\frac{7}{4}$ by $1$ .

$\frac{7}{4} (- 1 + \frac{7}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{7}{4} (- 1 \cdot \frac{4}{4} + \frac{7}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.3

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{7}{4} (\frac{- 1 \cdot 4}{4} + \frac{7}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.4

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot \frac{- 1 \cdot 4 + 7}{4} + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $- 1$ by $4$ .

$\frac{7}{4} \cdot \frac{- 4 + 7}{4} + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.3.2.5.2

Add $- 4$ and $7$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4

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

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Step 2.4.1

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

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (1 \cdot 1 - (- \frac{5}{4} \cdot - 1)) - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

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 $1$ by $1$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (1 - (- \frac{5}{4} \cdot - 1)) - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4.2.1.2

Multiply $- \frac{5}{4} \cdot - 1$ .

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Step 2.4.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (1 - (1 (\frac{5}{4}))) - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4.2.1.2.2

Multiply $\frac{5}{4}$ by $1$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (1 - \frac{5}{4}) - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4.2.2

Write $1$ as a fraction with a common denominator.

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (\frac{4}{4} - \frac{5}{4}) - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4.2.3

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} \cdot \frac{4 - 5}{4} - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4.2.4

Subtract $5$ from $4$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} \cdot \frac{- 1}{4} - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.4.2.5

Move the negative in front of the fraction.

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 2.5

Evaluate $| \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$ .

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Step 2.5.1

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

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 (1 (\frac{7}{4}) - (- \frac{5}{4} \cdot - 1))$

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 $\frac{7}{4}$ by $1$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{7}{4} - (- \frac{5}{4} \cdot - 1))$

Step 2.5.2.1.2

Multiply $- \frac{5}{4} \cdot - 1$ .

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Step 2.5.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{7}{4} - (1 (\frac{5}{4})))$

Step 2.5.2.1.2.2

Multiply $\frac{5}{4}$ by $1$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{7}{4} - \frac{5}{4})$

Step 2.5.2.2

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 \frac{7 - 5}{4}$

Step 2.5.2.3

Subtract $5$ from $7$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{2}{4})$

Step 2.5.2.4

Cancel the common factor of $2$ and $4$ .

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Step 2.5.2.4.1

Factor $2$ out of $2$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 \frac{2 (1)}{4}$

Step 2.5.2.4.2

Cancel the common factors.

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Step 2.5.2.4.2.1

Factor $2$ out of $4$ .

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 \frac{2 \cdot 1}{2 \cdot 2}$

Step 2.5.2.4.2.2

Cancel the common factor.

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 \frac{2 \cdot 1}{2 \cdot 2}$

Step 2.5.2.4.2.3

Rewrite the expression.

$\frac{7}{4} \cdot \frac{3}{4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{1}{2})$

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 $\frac{7}{4} \cdot \frac{3}{4}$ .

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Step 2.6.1.1.1

Multiply $\frac{7}{4}$ by $\frac{3}{4}$ .

$\frac{7 \cdot 3}{4 \cdot 4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{1}{2})$

Step 2.6.1.1.2

Multiply $7$ by $3$ .

$\frac{21}{4 \cdot 4} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{1}{2})$

Step 2.6.1.1.3

Multiply $4$ by $4$ .

$\frac{21}{16} + \frac{9}{4} (- \frac{1}{4}) - 1 (\frac{1}{2})$

Step 2.6.1.2

Multiply $\frac{9}{4} (- \frac{1}{4})$ .

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Step 2.6.1.2.1

Multiply $\frac{9}{4}$ by $\frac{1}{4}$ .

$\frac{21}{16} - \frac{9}{4 \cdot 4} - 1 (\frac{1}{2})$

Step 2.6.1.2.2

Multiply $4$ by $4$ .

$\frac{21}{16} - \frac{9}{16} - 1 (\frac{1}{2})$

Step 2.6.1.3

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\frac{21}{16} - \frac{9}{16} - \frac{1}{2}$

Step 2.6.2

Combine the numerators over the common denominator.

$\frac{21 - 9}{16} + \frac{- 1}{2}$

Step 2.6.3

Subtract $9$ from $21$ .

$\frac{12}{16} + \frac{- 1}{2}$

Step 2.6.4

Simplify each term.

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Step 2.6.4.1

Cancel the common factor of $12$ and $16$ .

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Step 2.6.4.1.1

Factor $4$ out of $12$ .

$\frac{4 (3)}{16} + \frac{- 1}{2}$

Step 2.6.4.1.2

Cancel the common factors.

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Step 2.6.4.1.2.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 3}{4 \cdot 4} + \frac{- 1}{2}$

Step 2.6.4.1.2.2

Cancel the common factor.

$\frac{4 \cdot 3}{4 \cdot 4} + \frac{- 1}{2}$

Step 2.6.4.1.2.3

Rewrite the expression.

$\frac{3}{4} + \frac{- 1}{2}$

Step 2.6.4.2

Move the negative in front of the fraction.

$\frac{3}{4} - \frac{1}{2}$

Step 2.6.5

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{4} - \frac{1}{2} \cdot \frac{2}{2}$

Step 2.6.6

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.6.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{3}{4} - \frac{2}{2 \cdot 2}$

Step 2.6.6.2

Multiply $2$ by $2$ .

$\frac{3}{4} - \frac{2}{4}$

Step 2.6.7

Combine the numerators over the common denominator.

$\frac{3 - 2}{4}$

Step 2.6.8

Subtract $2$ from $3$ .

$\frac{1}{4}$

$D = \frac{1}{4}$

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 $[\begin{array}{c} - 6 \\ - 5 \\ 1 \end{array}]$ .

$| \begin{array}{c} - 6 & - \frac{9}{4} & - 1 \\ - 5 & - 1 & - 1 \\ 1 & \frac{7}{4} & 1 \end{array} |$

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.

$| \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} |$

Step 4.2.1.4

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

$- 6 | \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} |$

Step 4.2.1.5

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

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

Step 4.2.1.6

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

$\frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} |$

Step 4.2.1.7

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

$| \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.1.8

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

$- 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.1.9

Add the terms together.

$- 6 | \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} | + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2

Evaluate $| \begin{array}{c} - 1 & - 1 \\ \frac{7}{4} & 1 \end{array} |$ .

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Step 4.2.2.1

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

$- 6 (- 1 \cdot 1 - \frac{7}{4} \cdot - 1) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2

Simplify the determinant.

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Step 4.2.2.2.1

Simplify each term.

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Step 4.2.2.2.1.1

Multiply $- 1$ by $1$ .

$- 6 (- 1 - \frac{7}{4} \cdot - 1) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.1.2

Multiply $- \frac{7}{4} \cdot - 1$ .

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Step 4.2.2.2.1.2.1

Multiply $- 1$ by $- 1$ .

$- 6 (- 1 + 1 (\frac{7}{4})) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.1.2.2

Multiply $\frac{7}{4}$ by $1$ .

$- 6 (- 1 + \frac{7}{4}) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 6 (- 1 \cdot \frac{4}{4} + \frac{7}{4}) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.3

Combine $- 1$ and $\frac{4}{4}$ .

$- 6 (\frac{- 1 \cdot 4}{4} + \frac{7}{4}) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.4

Combine the numerators over the common denominator.

$- 6 \frac{- 1 \cdot 4 + 7}{4} + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.5

Simplify the numerator.

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Step 4.2.2.2.5.1

Multiply $- 1$ by $4$ .

$- 6 \frac{- 4 + 7}{4} + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.2.2.5.2

Add $- 4$ and $7$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.3

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

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Step 4.2.3.1

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

$- 6 (\frac{3}{4}) + \frac{9}{4} (- 5 \cdot 1 - 1 \cdot - 1) - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.3.2

Simplify the determinant.

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Step 4.2.3.2.1

Simplify each term.

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Step 4.2.3.2.1.1

Multiply $- 5$ by $1$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} (- 5 - 1 \cdot - 1) - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.3.2.1.2

Multiply $- 1$ by $- 1$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} (- 5 + 1) - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.3.2.2

Add $- 5$ and $1$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 | \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$

Step 4.2.4

Evaluate $| \begin{array}{c} - 5 & - 1 \\ 1 & \frac{7}{4} \end{array} |$ .

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Step 4.2.4.1

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

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (- 5 (\frac{7}{4}) - 1 \cdot - 1)$

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 $- 5 (\frac{7}{4})$ .

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Step 4.2.4.2.1.1.1

Combine $- 5$ and $\frac{7}{4}$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (\frac{- 5 \cdot 7}{4} - 1 \cdot - 1)$

Step 4.2.4.2.1.1.2

Multiply $- 5$ by $7$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (\frac{- 35}{4} - 1 \cdot - 1)$

Step 4.2.4.2.1.2

Move the negative in front of the fraction.

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (- \frac{35}{4} - 1 \cdot - 1)$

Step 4.2.4.2.1.3

Multiply $- 1$ by $- 1$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (- \frac{35}{4} + 1)$

Step 4.2.4.2.2

Write $1$ as a fraction with a common denominator.

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (- \frac{35}{4} + \frac{4}{4})$

Step 4.2.4.2.3

Combine the numerators over the common denominator.

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 \frac{- 35 + 4}{4}$

Step 4.2.4.2.4

Add $- 35$ and $4$ .

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (\frac{- 31}{4})$

Step 4.2.4.2.5

Move the negative in front of the fraction.

$- 6 (\frac{3}{4}) + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5

Simplify the determinant.

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Step 4.2.5.1

Simplify each term.

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Step 4.2.5.1.1

Cancel the common factor of $2$ .

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Step 4.2.5.1.1.1

Factor $2$ out of $- 6$ .

$2 (- 3) \frac{3}{4} + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.1.2

Factor $2$ out of $4$ .

$2 \cdot - 3 \frac{3}{2 \cdot 2} + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.1.3

Cancel the common factor.

$2 \cdot - 3 \frac{3}{2 \cdot 2} + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.1.4

Rewrite the expression.

$- 3 (\frac{3}{2}) + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.2

Combine $- 3$ and $\frac{3}{2}$ .

$\frac{- 3 \cdot 3}{2} + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.3

Multiply $- 3$ by $3$ .

$\frac{- 9}{2} + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.4

Move the negative in front of the fraction.

$- \frac{9}{2} + \frac{9}{4} \cdot - 4 - 1 (- \frac{31}{4})$

Step 4.2.5.1.5

Cancel the common factor of $4$ .

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Step 4.2.5.1.5.1

Factor $4$ out of $- 4$ .

$- \frac{9}{2} + \frac{9}{4} \cdot (4 (- 1)) - 1 (- \frac{31}{4})$

Step 4.2.5.1.5.2

Cancel the common factor.

$- \frac{9}{2} + \frac{9}{4} \cdot (4 \cdot - 1) - 1 (- \frac{31}{4})$

Step 4.2.5.1.5.3

Rewrite the expression.

$- \frac{9}{2} + 9 \cdot - 1 - 1 (- \frac{31}{4})$

Step 4.2.5.1.6

Multiply $9$ by $- 1$ .

$- \frac{9}{2} - 9 - 1 (- \frac{31}{4})$

Step 4.2.5.1.7

Multiply $- 1 (- \frac{31}{4})$ .

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Step 4.2.5.1.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{9}{2} - 9 + 1 (\frac{31}{4})$

Step 4.2.5.1.7.2

Multiply $\frac{31}{4}$ by $1$ .

$- \frac{9}{2} - 9 + \frac{31}{4}$

Step 4.2.5.2

Find the common denominator.

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Step 4.2.5.2.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$- (\frac{9}{2} \cdot \frac{2}{2}) - 9 + \frac{31}{4}$

Step 4.2.5.2.2

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$- \frac{9 \cdot 2}{2 \cdot 2} - 9 + \frac{31}{4}$

Step 4.2.5.2.3

Write $- 9$ as a fraction with denominator $1$ .

$- \frac{9 \cdot 2}{2 \cdot 2} + \frac{- 9}{1} + \frac{31}{4}$

Step 4.2.5.2.4

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$- \frac{9 \cdot 2}{2 \cdot 2} + \frac{- 9}{1} \cdot \frac{4}{4} + \frac{31}{4}$

Step 4.2.5.2.5

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$- \frac{9 \cdot 2}{2 \cdot 2} + \frac{- 9 \cdot 4}{4} + \frac{31}{4}$

Step 4.2.5.2.6

Multiply $2$ by $2$ .

$- \frac{9 \cdot 2}{4} + \frac{- 9 \cdot 4}{4} + \frac{31}{4}$

Step 4.2.5.3

Combine the numerators over the common denominator.

$\frac{- 9 \cdot 2 - 9 \cdot 4 + 31}{4}$

Step 4.2.5.4

Simplify each term.

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Step 4.2.5.4.1

Multiply $- 9$ by $2$ .

$\frac{- 18 - 9 \cdot 4 + 31}{4}$

Step 4.2.5.4.2

Multiply $- 9$ by $4$ .

$\frac{- 18 - 36 + 31}{4}$

Step 4.2.5.5

Subtract $36$ from $- 18$ .

$\frac{- 54 + 31}{4}$

Step 4.2.5.6

Add $- 54$ and $31$ .

$\frac{- 23}{4}$

Step 4.2.5.7

Move the negative in front of the fraction.

$- \frac{23}{4}$

$D_{x} = - \frac{23}{4}$

Step 4.3

Use the formula to solve for $x$ .

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

Step 4.4

Substitute $\frac{1}{4}$ for $D$ and $- \frac{23}{4}$ for $D_{x}$ in the formula.

$x = \frac{- \frac{23}{4}}{\frac{1}{4}}$

Step 4.5

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{23}{4} \cdot 4$

Step 4.6

Cancel the common factor of $4$ .

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Step 4.6.1

Move the leading negative in $- \frac{23}{4}$ into the numerator.

$x = \frac{- 23}{4} \cdot 4$

Step 4.6.2

Cancel the common factor.

$x = \frac{- 23}{4} \cdot 4$

Step 4.6.3

Rewrite the expression.

$x = - 23$

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 $[\begin{array}{c} - 6 \\ - 5 \\ 1 \end{array}]$ .

$| \begin{array}{c} \frac{7}{4} & - 6 & - 1 \\ 1 & - 5 & - 1 \\ - \frac{5}{4} & 1 & 1 \end{array} |$

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.

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

Step 5.2.1.4

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

$\frac{7}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} |$

Step 5.2.1.5

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

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

Step 5.2.1.6

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

$6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.1.7

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

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

Step 5.2.1.8

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

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

Step 5.2.1.9

Add the terms together.

$\frac{7}{4} | \begin{array}{c} - 5 & - 1 \\ 1 & 1 \end{array} | + 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.2

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

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Step 5.2.2.1

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

$\frac{7}{4} (- 5 \cdot 1 - 1 \cdot - 1) + 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.2.2

Simplify the determinant.

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Step 5.2.2.2.1

Simplify each term.

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Step 5.2.2.2.1.1

Multiply $- 5$ by $1$ .

$\frac{7}{4} (- 5 - 1 \cdot - 1) + 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.2.2.1.2

Multiply $- 1$ by $- 1$ .

$\frac{7}{4} (- 5 + 1) + 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.2.2.2

Add $- 5$ and $1$ .

$\frac{7}{4} \cdot - 4 + 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & 1 \end{array} | - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3

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

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Step 5.2.3.1

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

$\frac{7}{4} \cdot - 4 + 6 (1 \cdot 1 - (- \frac{5}{4} \cdot - 1)) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

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 $1$ by $1$ .

$\frac{7}{4} \cdot - 4 + 6 (1 - (- \frac{5}{4} \cdot - 1)) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3.2.1.2

Multiply $- \frac{5}{4} \cdot - 1$ .

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Step 5.2.3.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{7}{4} \cdot - 4 + 6 (1 - (1 (\frac{5}{4}))) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3.2.1.2.2

Multiply $\frac{5}{4}$ by $1$ .

$\frac{7}{4} \cdot - 4 + 6 (1 - \frac{5}{4}) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3.2.2

Write $1$ as a fraction with a common denominator.

$\frac{7}{4} \cdot - 4 + 6 (\frac{4}{4} - \frac{5}{4}) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3.2.3

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot - 4 + 6 \frac{4 - 5}{4} - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3.2.4

Subtract $5$ from $4$ .

$\frac{7}{4} \cdot - 4 + 6 (\frac{- 1}{4}) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.3.2.5

Move the negative in front of the fraction.

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 5.2.4

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

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Step 5.2.4.1

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

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (1 \cdot 1 - (- \frac{5}{4} \cdot - 5))$

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 $1$ by $1$ .

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (1 - (- \frac{5}{4} \cdot - 5))$

Step 5.2.4.2.1.2

Multiply $- \frac{5}{4} \cdot - 5$ .

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Step 5.2.4.2.1.2.1

Multiply $- 5$ by $- 1$ .

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (1 - (5 (\frac{5}{4})))$

Step 5.2.4.2.1.2.2

Combine $5$ and $\frac{5}{4}$ .

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (1 - \frac{5 \cdot 5}{4})$

Step 5.2.4.2.1.2.3

Multiply $5$ by $5$ .

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (1 - \frac{25}{4})$

Step 5.2.4.2.2

Write $1$ as a fraction with a common denominator.

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (\frac{4}{4} - \frac{25}{4})$

Step 5.2.4.2.3

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 \frac{4 - 25}{4}$

Step 5.2.4.2.4

Subtract $25$ from $4$ .

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (\frac{- 21}{4})$

Step 5.2.4.2.5

Move the negative in front of the fraction.

$\frac{7}{4} \cdot - 4 + 6 (- \frac{1}{4}) - 1 (- \frac{21}{4})$

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

Cancel the common factor of $4$ .

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Step 5.2.5.1.1.1

Factor $4$ out of $- 4$ .

$\frac{7}{4} \cdot (4 (- 1)) + 6 (- \frac{1}{4}) - 1 (- \frac{21}{4})$

Step 5.2.5.1.1.2

Cancel the common factor.

$\frac{7}{4} \cdot (4 \cdot - 1) + 6 (- \frac{1}{4}) - 1 (- \frac{21}{4})$

Step 5.2.5.1.1.3

Rewrite the expression.

$7 \cdot - 1 + 6 (- \frac{1}{4}) - 1 (- \frac{21}{4})$

Step 5.2.5.1.2

Multiply $7$ by $- 1$ .

$- 7 + 6 (- \frac{1}{4}) - 1 (- \frac{21}{4})$

Step 5.2.5.1.3

Cancel the common factor of $2$ .

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Step 5.2.5.1.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$- 7 + 6 (\frac{- 1}{4}) - 1 (- \frac{21}{4})$

Step 5.2.5.1.3.2

Factor $2$ out of $6$ .

$- 7 + 2 (3) \frac{- 1}{4} - 1 (- \frac{21}{4})$

Step 5.2.5.1.3.3

Factor $2$ out of $4$ .

$- 7 + 2 \cdot 3 \frac{- 1}{2 \cdot 2} - 1 (- \frac{21}{4})$

Step 5.2.5.1.3.4

Cancel the common factor.

$- 7 + 2 \cdot 3 \frac{- 1}{2 \cdot 2} - 1 (- \frac{21}{4})$

Step 5.2.5.1.3.5

Rewrite the expression.

$- 7 + 3 (\frac{- 1}{2}) - 1 (- \frac{21}{4})$

Step 5.2.5.1.4

Combine $3$ and $\frac{- 1}{2}$ .

$- 7 + \frac{3 \cdot - 1}{2} - 1 (- \frac{21}{4})$

Step 5.2.5.1.5

Multiply $3$ by $- 1$ .

$- 7 + \frac{- 3}{2} - 1 (- \frac{21}{4})$

Step 5.2.5.1.6

Move the negative in front of the fraction.

$- 7 - \frac{3}{2} - 1 (- \frac{21}{4})$

Step 5.2.5.1.7

Multiply $- 1 (- \frac{21}{4})$ .

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Step 5.2.5.1.7.1

Multiply $- 1$ by $- 1$ .

$- 7 - \frac{3}{2} + 1 (\frac{21}{4})$

Step 5.2.5.1.7.2

Multiply $\frac{21}{4}$ by $1$ .

$- 7 - \frac{3}{2} + \frac{21}{4}$

Step 5.2.5.2

Find the common denominator.

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Step 5.2.5.2.1

Write $- 7$ as a fraction with denominator $1$ .

$\frac{- 7}{1} - \frac{3}{2} + \frac{21}{4}$

Step 5.2.5.2.2

Multiply $\frac{- 7}{1}$ by $\frac{4}{4}$ .

$\frac{- 7}{1} \cdot \frac{4}{4} - \frac{3}{2} + \frac{21}{4}$

Step 5.2.5.2.3

Multiply $\frac{- 7}{1}$ by $\frac{4}{4}$ .

$\frac{- 7 \cdot 4}{4} - \frac{3}{2} + \frac{21}{4}$

Step 5.2.5.2.4

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{- 7 \cdot 4}{4} - (\frac{3}{2} \cdot \frac{2}{2}) + \frac{21}{4}$

Step 5.2.5.2.5

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{- 7 \cdot 4}{4} - \frac{3 \cdot 2}{2 \cdot 2} + \frac{21}{4}$

Step 5.2.5.2.6

Multiply $2$ by $2$ .

$\frac{- 7 \cdot 4}{4} - \frac{3 \cdot 2}{4} + \frac{21}{4}$

Step 5.2.5.3

Combine the numerators over the common denominator.

$\frac{- 7 \cdot 4 - 3 \cdot 2 + 21}{4}$

Step 5.2.5.4

Simplify each term.

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Step 5.2.5.4.1

Multiply $- 7$ by $4$ .

$\frac{- 28 - 3 \cdot 2 + 21}{4}$

Step 5.2.5.4.2

Multiply $- 3$ by $2$ .

$\frac{- 28 - 6 + 21}{4}$

Step 5.2.5.5

Subtract $6$ from $- 28$ .

$\frac{- 34 + 21}{4}$

Step 5.2.5.6

Add $- 34$ and $21$ .

$\frac{- 13}{4}$

Step 5.2.5.7

Move the negative in front of the fraction.

$- \frac{13}{4}$

$D_{y} = - \frac{13}{4}$

Step 5.3

Use the formula to solve for $y$ .

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

Step 5.4

Substitute $\frac{1}{4}$ for $D$ and $- \frac{13}{4}$ for $D_{y}$ in the formula.

$y = \frac{- \frac{13}{4}}{\frac{1}{4}}$

Step 5.5

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{13}{4} \cdot 4$

Step 5.6

Cancel the common factor of $4$ .

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Step 5.6.1

Move the leading negative in $- \frac{13}{4}$ into the numerator.

$y = \frac{- 13}{4} \cdot 4$

Step 5.6.2

Cancel the common factor.

$y = \frac{- 13}{4} \cdot 4$

Step 5.6.3

Rewrite the expression.

$y = - 13$

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 $[\begin{array}{c} - 6 \\ - 5 \\ 1 \end{array}]$ .

$| \begin{array}{c} \frac{7}{4} & - \frac{9}{4} & - 6 \\ 1 & - 1 & - 5 \\ - \frac{5}{4} & \frac{7}{4} & 1 \end{array} |$

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.

$| \begin{array}{c} - 1 & - 5 \\ \frac{7}{4} & 1 \end{array} |$

Step 6.2.1.4

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

$\frac{7}{4} | \begin{array}{c} - 1 & - 5 \\ \frac{7}{4} & 1 \end{array} |$

Step 6.2.1.5

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

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

Step 6.2.1.6

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

$\frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} |$

Step 6.2.1.7

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

$| \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.1.8

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

$- 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.1.9

Add the terms together.

$\frac{7}{4} | \begin{array}{c} - 1 & - 5 \\ \frac{7}{4} & 1 \end{array} | + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2

Evaluate $| \begin{array}{c} - 1 & - 5 \\ \frac{7}{4} & 1 \end{array} |$ .

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Step 6.2.2.1

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

$\frac{7}{4} (- 1 \cdot 1 - \frac{7}{4} \cdot - 5) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2

Simplify the determinant.

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Step 6.2.2.2.1

Simplify each term.

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Step 6.2.2.2.1.1

Multiply $- 1$ by $1$ .

$\frac{7}{4} (- 1 - \frac{7}{4} \cdot - 5) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.1.2

Multiply $- \frac{7}{4} \cdot - 5$ .

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Step 6.2.2.2.1.2.1

Multiply $- 5$ by $- 1$ .

$\frac{7}{4} (- 1 + 5 (\frac{7}{4})) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.1.2.2

Combine $5$ and $\frac{7}{4}$ .

$\frac{7}{4} (- 1 + \frac{5 \cdot 7}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.1.2.3

Multiply $5$ by $7$ .

$\frac{7}{4} (- 1 + \frac{35}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{7}{4} (- 1 \cdot \frac{4}{4} + \frac{35}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.3

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{7}{4} (\frac{- 1 \cdot 4}{4} + \frac{35}{4}) + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.4

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot \frac{- 1 \cdot 4 + 35}{4} + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.5

Simplify the numerator.

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Step 6.2.2.2.5.1

Multiply $- 1$ by $4$ .

$\frac{7}{4} \cdot \frac{- 4 + 35}{4} + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.2.2.5.2

Add $- 4$ and $35$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} | \begin{array}{c} 1 & - 5 \\ - \frac{5}{4} & 1 \end{array} | - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3

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

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Step 6.2.3.1

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

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (1 \cdot 1 - (- \frac{5}{4} \cdot - 5)) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

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 $1$ by $1$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (1 - (- \frac{5}{4} \cdot - 5)) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.1.2

Multiply $- \frac{5}{4} \cdot - 5$ .

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Step 6.2.3.2.1.2.1

Multiply $- 5$ by $- 1$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (1 - (5 (\frac{5}{4}))) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.1.2.2

Combine $5$ and $\frac{5}{4}$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (1 - \frac{5 \cdot 5}{4}) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.1.2.3

Multiply $5$ by $5$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (1 - \frac{25}{4}) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.2

Write $1$ as a fraction with a common denominator.

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (\frac{4}{4} - \frac{25}{4}) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.3

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} \cdot \frac{4 - 25}{4} - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.4

Subtract $25$ from $4$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} \cdot \frac{- 21}{4} - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.3.2.5

Move the negative in front of the fraction.

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 | \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$

Step 6.2.4

Evaluate $| \begin{array}{c} 1 & - 1 \\ - \frac{5}{4} & \frac{7}{4} \end{array} |$ .

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Step 6.2.4.1

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

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 (1 (\frac{7}{4}) - (- \frac{5}{4} \cdot - 1))$

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 $\frac{7}{4}$ by $1$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{7}{4} - (- \frac{5}{4} \cdot - 1))$

Step 6.2.4.2.1.2

Multiply $- \frac{5}{4} \cdot - 1$ .

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Step 6.2.4.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{7}{4} - (1 (\frac{5}{4})))$

Step 6.2.4.2.1.2.2

Multiply $\frac{5}{4}$ by $1$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{7}{4} - \frac{5}{4})$

Step 6.2.4.2.2

Combine the numerators over the common denominator.

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 \frac{7 - 5}{4}$

Step 6.2.4.2.3

Subtract $5$ from $7$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{2}{4})$

Step 6.2.4.2.4

Cancel the common factor of $2$ and $4$ .

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Step 6.2.4.2.4.1

Factor $2$ out of $2$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 \frac{2 (1)}{4}$

Step 6.2.4.2.4.2

Cancel the common factors.

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Step 6.2.4.2.4.2.1

Factor $2$ out of $4$ .

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 \frac{2 \cdot 1}{2 \cdot 2}$

Step 6.2.4.2.4.2.2

Cancel the common factor.

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 \frac{2 \cdot 1}{2 \cdot 2}$

Step 6.2.4.2.4.2.3

Rewrite the expression.

$\frac{7}{4} \cdot \frac{31}{4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{1}{2})$

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 $\frac{7}{4} \cdot \frac{31}{4}$ .

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Step 6.2.5.1.1.1

Multiply $\frac{7}{4}$ by $\frac{31}{4}$ .

$\frac{7 \cdot 31}{4 \cdot 4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{1}{2})$

Step 6.2.5.1.1.2

Multiply $7$ by $31$ .

$\frac{217}{4 \cdot 4} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{1}{2})$

Step 6.2.5.1.1.3

Multiply $4$ by $4$ .

$\frac{217}{16} + \frac{9}{4} (- \frac{21}{4}) - 6 (\frac{1}{2})$

Step 6.2.5.1.2

Multiply $\frac{9}{4} (- \frac{21}{4})$ .

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Step 6.2.5.1.2.1

Multiply $\frac{9}{4}$ by $\frac{21}{4}$ .

$\frac{217}{16} - \frac{9 \cdot 21}{4 \cdot 4} - 6 (\frac{1}{2})$

Step 6.2.5.1.2.2

Multiply $9$ by $21$ .

$\frac{217}{16} - \frac{189}{4 \cdot 4} - 6 (\frac{1}{2})$

Step 6.2.5.1.2.3

Multiply $4$ by $4$ .

$\frac{217}{16} - \frac{189}{16} - 6 (\frac{1}{2})$

Step 6.2.5.1.3

Cancel the common factor of $2$ .

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Step 6.2.5.1.3.1

Factor $2$ out of $- 6$ .

$\frac{217}{16} - \frac{189}{16} + 2 (- 3) \frac{1}{2}$

Step 6.2.5.1.3.2

Cancel the common factor.

$\frac{217}{16} - \frac{189}{16} + 2 \cdot - 3 \frac{1}{2}$

Step 6.2.5.1.3.3

Rewrite the expression.

$\frac{217}{16} - \frac{189}{16} - 3$

Step 6.2.5.2

Combine the numerators over the common denominator.

$- 3 + \frac{217 - 189}{16}$

Step 6.2.5.3

Subtract $189$ from $217$ .

$- 3 + \frac{28}{16}$

Step 6.2.5.4

Cancel the common factor of $28$ and $16$ .

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Step 6.2.5.4.1

Factor $4$ out of $28$ .

$- 3 + \frac{4 (7)}{16}$

Step 6.2.5.4.2

Cancel the common factors.

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Step 6.2.5.4.2.1

Factor $4$ out of $16$ .

$- 3 + \frac{4 \cdot 7}{4 \cdot 4}$

Step 6.2.5.4.2.2

Cancel the common factor.

$- 3 + \frac{4 \cdot 7}{4 \cdot 4}$

Step 6.2.5.4.2.3

Rewrite the expression.

$- 3 + \frac{7}{4}$

Step 6.2.5.5

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 3 \cdot \frac{4}{4} + \frac{7}{4}$

Step 6.2.5.6

Combine $- 3$ and $\frac{4}{4}$ .

$\frac{- 3 \cdot 4}{4} + \frac{7}{4}$

Step 6.2.5.7

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 4 + 7}{4}$

Step 6.2.5.8

Simplify the numerator.

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Step 6.2.5.8.1

Multiply $- 3$ by $4$ .

$\frac{- 12 + 7}{4}$

Step 6.2.5.8.2

Add $- 12$ and $7$ .

$\frac{- 5}{4}$

Step 6.2.5.9

Move the negative in front of the fraction.

$- \frac{5}{4}$

$D_{z} = - \frac{5}{4}$

Step 6.3

Use the formula to solve for $z$ .

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

Step 6.4

Substitute $\frac{1}{4}$ for $D$ and $- \frac{5}{4}$ for $D_{z}$ in the formula.

$z = \frac{- \frac{5}{4}}{\frac{1}{4}}$

Step 6.5

Multiply the numerator by the reciprocal of the denominator.

$z = - \frac{5}{4} \cdot 4$

Step 6.6

Cancel the common factor of $4$ .

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Step 6.6.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$z = \frac{- 5}{4} \cdot 4$

Step 6.6.2

Cancel the common factor.

$z = \frac{- 5}{4} \cdot 4$

Step 6.6.3

Rewrite the expression.

$z = - 5$

Step 7

List the solution to the system of equations.

$x = - 23$

$y = - 13$

$z = - 5$