Finite Math Examples

$x + 5 z = - 6 + x$ , $3 x + 3 y = 10 + z$ , $x + 3 y + 2 z = 5$

Step 1

Solve for $z$ in $x + 5 z = - 6 + x$ .

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Step 1.1

Move all terms not containing $z$ to the right side of the equation.

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$5 z = - 6 + x - x$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 1.1.2

Combine the opposite terms in $- 6 + x - x$ .

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Step 1.1.2.1

Subtract $x$ from $x$ .

$5 z = - 6 + 0$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 1.1.2.2

Add $- 6$ and $0$ .

$5 z = - 6$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$5 z = - 6$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$5 z = - 6$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 1.2

Divide each term in $5 z = - 6$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 z = - 6$ by $5$ .

$\frac{5 z}{5} = \frac{- 6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 z}{5} = \frac{- 6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 1.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$z = \frac{- 6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$z = \frac{- 6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$z = - \frac{6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$z = - \frac{6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$z = - \frac{6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

$z = - \frac{6}{5}$

$3 x + 3 y = 10 + z$

$x + 3 y + 2 z = 5$

Step 2

Replace all occurrences of $z$ with $- \frac{6}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $3 x + 3 y = 10 + z$ with $- \frac{6}{5}$ .

$3 x + 3 y = 10 - \frac{6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.2

Simplify $3 x + 3 y = 10 - \frac{6}{5}$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Remove parentheses.

$3 x + 3 y = 10 - \frac{6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

$3 x + 3 y = 10 - \frac{6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $10 - \frac{6}{5}$ .

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Step 2.2.2.1.1

To write $10$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3 x + 3 y = 10 \cdot \frac{5}{5} - \frac{6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.2.2.1.2

Combine $10$ and $\frac{5}{5}$ .

$3 x + 3 y = \frac{10 \cdot 5}{5} - \frac{6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.2.2.1.3

Combine the numerators over the common denominator.

$3 x + 3 y = \frac{10 \cdot 5 - 6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.2.2.1.4

Simplify the numerator.

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Step 2.2.2.1.4.1

Multiply $10$ by $5$ .

$3 x + 3 y = \frac{50 - 6}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.2.2.1.4.2

Subtract $6$ from $50$ .

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y + 2 z = 5$

Step 2.3

Replace all occurrences of $z$ in $x + 3 y + 2 z = 5$ with $- \frac{6}{5}$ .

$x + 3 y + 2 (- \frac{6}{5}) = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $2 (- \frac{6}{5})$ .

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Step 2.4.1.1.1

Multiply $- 1$ by $2$ .

$x + 3 y - 2 (\frac{6}{5}) = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 2.4.1.1.2

Combine $- 2$ and $\frac{6}{5}$ .

$x + 3 y + \frac{- 2 \cdot 6}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 2.4.1.1.3

Multiply $- 2$ by $6$ .

$x + 3 y + \frac{- 12}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y + \frac{- 12}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 2.4.1.2

Move the negative in front of the fraction.

$x + 3 y - \frac{12}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y - \frac{12}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y - \frac{12}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x + 3 y - \frac{12}{5} = 5$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $3 y$ from both sides of the equation.

$x - \frac{12}{5} = 5 - 3 y$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3.2

Add $\frac{12}{5}$ to both sides of the equation.

$x = 5 - 3 y + \frac{12}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - 3 y + 5 \cdot \frac{5}{5} + \frac{12}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3.4

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

$x = - 3 y + \frac{5 \cdot 5}{5} + \frac{12}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3.5

Combine the numerators over the common denominator.

$x = - 3 y + \frac{5 \cdot 5 + 12}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $5$ by $5$ .

$x = - 3 y + \frac{25 + 12}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 3.6.2

Add $25$ and $12$ .

$x = - 3 y + \frac{37}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x = - 3 y + \frac{37}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

$x = - 3 y + \frac{37}{5}$

$3 x + 3 y = \frac{44}{5}$

$z = - \frac{6}{5}$

Step 4

Replace all occurrences of $x$ with $- 3 y + \frac{37}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $3 x + 3 y = \frac{44}{5}$ with $- 3 y + \frac{37}{5}$ .

$3 (- 3 y + \frac{37}{5}) + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $3 (- 3 y + \frac{37}{5}) + 3 y$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$3 (- 3 y) + 3 (\frac{37}{5}) + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 4.2.1.1.2

Multiply $- 3$ by $3$ .

$- 9 y + 3 (\frac{37}{5}) + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 4.2.1.1.3

Multiply $3 (\frac{37}{5})$ .

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Step 4.2.1.1.3.1

Combine $3$ and $\frac{37}{5}$ .

$- 9 y + \frac{3 \cdot 37}{5} + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 4.2.1.1.3.2

Multiply $3$ by $37$ .

$- 9 y + \frac{111}{5} + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$- 9 y + \frac{111}{5} + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$- 9 y + \frac{111}{5} + 3 y = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 4.2.1.2

Add $- 9 y$ and $3 y$ .

$- 6 y + \frac{111}{5} = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$- 6 y + \frac{111}{5} = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$- 6 y + \frac{111}{5} = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$- 6 y + \frac{111}{5} = \frac{44}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5

Solve for $y$ in $- 6 y + \frac{111}{5} = \frac{44}{5}$ .

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Step 5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 5.1.1

Subtract $\frac{111}{5}$ from both sides of the equation.

$- 6 y = \frac{44}{5} - \frac{111}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.1.2

Combine the numerators over the common denominator.

$- 6 y = \frac{44 - 111}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.1.3

Subtract $111$ from $44$ .

$- 6 y = \frac{- 67}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.1.4

Move the negative in front of the fraction.

$- 6 y = - \frac{67}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$- 6 y = - \frac{67}{5}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2

Divide each term in $- 6 y = - \frac{67}{5}$ by $- 6$ and simplify.

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Step 5.2.1

Divide each term in $- 6 y = - \frac{67}{5}$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{- \frac{67}{5}}{- 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{- \frac{67}{5}}{- 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{67}{5}}{- 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$y = \frac{- \frac{67}{5}}{- 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$y = \frac{- \frac{67}{5}}{- 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{67}{5} \cdot \frac{1}{- 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.3.2

Move the negative in front of the fraction.

$y = - \frac{67}{5} \cdot (- \frac{1}{6})$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.3.3

Multiply $- \frac{67}{5} (- \frac{1}{6})$ .

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Step 5.2.3.3.1

Multiply $- 1$ by $- 1$ .

$y = 1 (\frac{67}{5}) \cdot \frac{1}{6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.3.3.2

Multiply $\frac{67}{5}$ by $1$ .

$y = \frac{67}{5} \cdot \frac{1}{6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.3.3.3

Multiply $\frac{67}{5}$ by $\frac{1}{6}$ .

$y = \frac{67}{5 \cdot 6}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 5.2.3.3.4

Multiply $5$ by $6$ .

$y = \frac{67}{30}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$y = \frac{67}{30}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$y = \frac{67}{30}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$y = \frac{67}{30}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

$y = \frac{67}{30}$

$x = - 3 y + \frac{37}{5}$

$z = - \frac{6}{5}$

Step 6

Replace all occurrences of $y$ with $\frac{67}{30}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = - 3 y + \frac{37}{5}$ with $\frac{67}{30}$ .

$x = - 3 (\frac{67}{30}) + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 3 (\frac{67}{30}) + \frac{37}{5}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1.1

Factor $3$ out of $- 3$ .

$x = 3 (- 1) (\frac{67}{30}) + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.1.1.2

Factor $3$ out of $30$ .

$x = 3 \cdot (- 1 \frac{67}{3 \cdot 10}) + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.1.1.3

Cancel the common factor.

$x = 3 \cdot (- 1 \frac{67}{3 \cdot 10}) + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.1.1.4

Rewrite the expression.

$x = - 1 (\frac{67}{10}) + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = - 1 (\frac{67}{10}) + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.1.2

Rewrite $- 1 (\frac{67}{10})$ as $- (\frac{67}{10})$ .

$x = - \frac{67}{10} + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = - \frac{67}{10} + \frac{37}{5}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.2

To write $\frac{37}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{67}{10} + \frac{37}{5} \cdot \frac{2}{2}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.1.3.1

Multiply $\frac{37}{5}$ by $\frac{2}{2}$ .

$x = - \frac{67}{10} + \frac{37 \cdot 2}{5 \cdot 2}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.3.2

Multiply $5$ by $2$ .

$x = - \frac{67}{10} + \frac{37 \cdot 2}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = - \frac{67}{10} + \frac{37 \cdot 2}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 67 + 37 \cdot 2}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $37$ by $2$ .

$x = \frac{- 67 + 74}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 6.2.1.5.2

Add $- 67$ and $74$ .

$x = \frac{7}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = \frac{7}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = \frac{7}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = \frac{7}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

$x = \frac{7}{10}$

$y = \frac{67}{30}$

$z = - \frac{6}{5}$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{7}{10}, \frac{67}{30}, - \frac{6}{5})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{7}{10}, \frac{67}{30}, - \frac{6}{5})$

Equation Form:

$x = \frac{7}{10}, y = \frac{67}{30}, z = - \frac{6}{5}$