Finite Math Examples

$x + y - z = 1$ , $x + 3 z = - 5$ , $3 y - z = 5$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x - z = 1 - y$

$x + 3 z = - 5$

$3 y - z = 5$

Step 1.2

Add $z$ to both sides of the equation.

$x = 1 - y + z$

$x + 3 z = - 5$

$3 y - z = 5$

$x = 1 - y + z$

$x + 3 z = - 5$

$3 y - z = 5$

Step 2

Replace all occurrences of $x$ with $1 - y + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x + 3 z = - 5$ with $1 - y + z$ .

$(1 - y + z) + 3 z = - 5$

$x = 1 - y + z$

$3 y - z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Add $z$ and $3 z$ .

$1 - y + 4 z = - 5$

$x = 1 - y + z$

$3 y - z = 5$

$1 - y + 4 z = - 5$

$x = 1 - y + z$

$3 y - z = 5$

$1 - y + 4 z = - 5$

$x = 1 - y + z$

$3 y - z = 5$

Step 3

Solve for $y$ in $1 - y + 4 z = - 5$ .

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Step 3.1

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

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Step 3.1.1

Subtract $1$ from both sides of the equation.

$- y + 4 z = - 5 - 1$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.1.2

Subtract $4 z$ from both sides of the equation.

$- y = - 5 - 1 - 4 z$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.1.3

Subtract $1$ from $- 5$ .

$- y = - 6 - 4 z$

$x = 1 - y + z$

$3 y - z = 5$

$- y = - 6 - 4 z$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2

Divide each term in $- y = - 6 - 4 z$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y = - 6 - 4 z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 6}{- 1} + \frac{- 4 z}{- 1}$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 6}{- 1} + \frac{- 4 z}{- 1}$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 6}{- 1} + \frac{- 4 z}{- 1}$

$x = 1 - y + z$

$3 y - z = 5$

$y = \frac{- 6}{- 1} + \frac{- 4 z}{- 1}$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 6$ by $- 1$ .

$y = 6 + \frac{- 4 z}{- 1}$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{- 4 z}{- 1}$ .

$y = 6 - 1 \cdot (- 4 z)$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2.3.1.3

Rewrite $- 1 \cdot (- 4 z)$ as $- (- 4 z)$ .

$y = 6 - (- 4 z)$

$x = 1 - y + z$

$3 y - z = 5$

Step 3.2.3.1.4

Multiply $- 4$ by $- 1$ .

$y = 6 + 4 z$

$x = 1 - y + z$

$3 y - z = 5$

$y = 6 + 4 z$

$x = 1 - y + z$

$3 y - z = 5$

$y = 6 + 4 z$

$x = 1 - y + z$

$3 y - z = 5$

$y = 6 + 4 z$

$x = 1 - y + z$

$3 y - z = 5$

$y = 6 + 4 z$

$x = 1 - y + z$

$3 y - z = 5$

Step 4

Replace all occurrences of $y$ with $6 + 4 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 1 - y + z$ with $6 + 4 z$ .

$x = 1 - (6 + 4 z) + z$

$y = 6 + 4 z$

$3 y - z = 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $1 - (6 + 4 z) + z$ .

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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.

$x = 1 - 1 \cdot 6 - (4 z) + z$

$y = 6 + 4 z$

$3 y - z = 5$

Step 4.2.1.1.2

Multiply $- 1$ by $6$ .

$x = 1 - 6 - (4 z) + z$

$y = 6 + 4 z$

$3 y - z = 5$

Step 4.2.1.1.3

Multiply $4$ by $- 1$ .

$x = 1 - 6 - 4 z + z$

$y = 6 + 4 z$

$3 y - z = 5$

$x = 1 - 6 - 4 z + z$

$y = 6 + 4 z$

$3 y - z = 5$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $6$ from $1$ .

$x = - 5 - 4 z + z$

$y = 6 + 4 z$

$3 y - z = 5$

Step 4.2.1.2.2

Add $- 4 z$ and $z$ .

$x = - 5 - 3 z$

$y = 6 + 4 z$

$3 y - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$3 y - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$3 y - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$3 y - z = 5$

Step 4.3

Replace all occurrences of $y$ in $3 y - z = 5$ with $6 + 4 z$ .

$3 (6 + 4 z) - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $3 (6 + 4 z) - z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$3 \cdot 6 + 3 (4 z) - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 4.4.1.1.2

Multiply $3$ by $6$ .

$18 + 3 (4 z) - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 4.4.1.1.3

Multiply $4$ by $3$ .

$18 + 12 z - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$18 + 12 z - z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 4.4.1.2

Subtract $z$ from $12 z$ .

$18 + 11 z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$18 + 11 z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$18 + 11 z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$18 + 11 z = 5$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 5

Solve for $z$ in $18 + 11 z = 5$ .

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Step 5.1

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

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Step 5.1.1

Subtract $18$ from both sides of the equation.

$11 z = 5 - 18$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 5.1.2

Subtract $18$ from $5$ .

$11 z = - 13$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$11 z = - 13$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 5.2

Divide each term in $11 z = - 13$ by $11$ and simplify.

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Step 5.2.1

Divide each term in $11 z = - 13$ by $11$ .

$\frac{11 z}{11} = \frac{- 13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $11$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{11 z}{11} = \frac{- 13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$z = \frac{- 13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$z = \frac{- 13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$z = - \frac{13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$z = - \frac{13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$z = - \frac{13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

$z = - \frac{13}{11}$

$x = - 5 - 3 z$

$y = 6 + 4 z$

Step 6

Replace all occurrences of $z$ with $- \frac{13}{11}$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = - 5 - 3 z$ with $- \frac{13}{11}$ .

$x = - 5 - 3 (- \frac{13}{11})$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 5 - 3 (- \frac{13}{11})$ .

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Step 6.2.1.1

Multiply $- 3 (- \frac{13}{11})$ .

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Step 6.2.1.1.1

Multiply $- 1$ by $- 3$ .

$x = - 5 + 3 (\frac{13}{11})$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.1.2

Combine $3$ and $\frac{13}{11}$ .

$x = - 5 + \frac{3 \cdot 13}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.1.3

Multiply $3$ by $13$ .

$x = - 5 + \frac{39}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

$x = - 5 + \frac{39}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x = - 5 \cdot \frac{11}{11} + \frac{39}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.3

Combine $- 5$ and $\frac{11}{11}$ .

$x = \frac{- 5 \cdot 11}{11} + \frac{39}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 5 \cdot 11 + 39}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $- 5$ by $11$ .

$x = \frac{- 55 + 39}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.5.2

Add $- 55$ and $39$ .

$x = \frac{- 16}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

$x = \frac{- 16}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.2.1.6

Move the negative in front of the fraction.

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = 6 + 4 z$

Step 6.3

Replace all occurrences of $z$ in $y = 6 + 4 z$ with $- \frac{13}{11}$ .

$y = 6 + 4 (- \frac{13}{11})$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $6 + 4 (- \frac{13}{11})$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Multiply $4 (- \frac{13}{11})$ .

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Step 6.4.1.1.1.1

Multiply $- 1$ by $4$ .

$y = 6 - 4 (\frac{13}{11})$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.1.1.2

Combine $- 4$ and $\frac{13}{11}$ .

$y = 6 + \frac{- 4 \cdot 13}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.1.1.3

Multiply $- 4$ by $13$ .

$y = 6 + \frac{- 52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = 6 + \frac{- 52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.1.2

Move the negative in front of the fraction.

$y = 6 - \frac{52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = 6 - \frac{52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = 6 \cdot \frac{11}{11} - \frac{52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.3

Combine $6$ and $\frac{11}{11}$ .

$y = \frac{6 \cdot 11}{11} - \frac{52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$y = \frac{6 \cdot 11 - 52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $6$ by $11$ .

$y = \frac{66 - 52}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 6.4.1.5.2

Subtract $52$ from $66$ .

$y = \frac{14}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = \frac{14}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = \frac{14}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = \frac{14}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

$y = \frac{14}{11}$

$x = - \frac{16}{11}$

$z = - \frac{13}{11}$

Step 7

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

$(- \frac{16}{11}, \frac{14}{11}, - \frac{13}{11})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- \frac{16}{11}, \frac{14}{11}, - \frac{13}{11})$

Equation Form:

$x = - \frac{16}{11}, y = \frac{14}{11}, z = - \frac{13}{11}$