Finite Math Examples

$x + 4 y - 3 z = - 8$ , $3 x - y + 3 z = 12$ , $x + y + 6 z = 1$

Step 1

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

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Step 1.1

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

$x - 3 z = - 8 - 4 y$

$3 x - y + 3 z = 12$

$x + y + 6 z = 1$

Step 1.2

Add $3 z$ to both sides of the equation.

$x = - 8 - 4 y + 3 z$

$3 x - y + 3 z = 12$

$x + y + 6 z = 1$

$x = - 8 - 4 y + 3 z$

$3 x - y + 3 z = 12$

$x + y + 6 z = 1$

Step 2

Replace all occurrences of $x$ with $- 8 - 4 y + 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - y + 3 z = 12$ with $- 8 - 4 y + 3 z$ .

$3 (- 8 - 4 y + 3 z) - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- 8 - 4 y + 3 z) - y + 3 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 \cdot - 8 + 3 (- 4 y) + 3 (3 z) - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $- 8$ .

$- 24 + 3 (- 4 y) + 3 (3 z) - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.2.1.1.2.2

Multiply $- 4$ by $3$ .

$- 24 - 12 y + 3 (3 z) - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.2.1.1.2.3

Multiply $3$ by $3$ .

$- 24 - 12 y + 9 z - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

$- 24 - 12 y + 9 z - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

$- 24 - 12 y + 9 z - y + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $y$ from $- 12 y$ .

$- 24 - 13 y + 9 z + 3 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.2.1.2.2

Add $9 z$ and $3 z$ .

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$x + y + 6 z = 1$

Step 2.3

Replace all occurrences of $x$ in $x + y + 6 z = 1$ with $- 8 - 4 y + 3 z$ .

$(- 8 - 4 y + 3 z) + y + 6 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $(- 8 - 4 y + 3 z) + y + 6 z$ .

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Step 2.4.1.1

Add $- 4 y$ and $y$ .

$- 8 - 3 y + 3 z + 6 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 2.4.1.2

Add $3 z$ and $6 z$ .

$- 8 - 3 y + 9 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$- 8 - 3 y + 9 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$- 8 - 3 y + 9 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$- 8 - 3 y + 9 z = 1$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3

Solve for $y$ in $- 8 - 3 y + 9 z = 1$ .

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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

Add $8$ to both sides of the equation.

$- 3 y + 9 z = 1 + 8$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.1.2

Subtract $9 z$ from both sides of the equation.

$- 3 y = 1 + 8 - 9 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.1.3

Add $1$ and $8$ .

$- 3 y = 9 - 9 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$- 3 y = 9 - 9 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2

Divide each term in $- 3 y = 9 - 9 z$ by $- 3$ and simplify.

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Step 3.2.1

Divide each term in $- 3 y = 9 - 9 z$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{9}{- 3} + \frac{- 9 z}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{9}{- 3} + \frac{- 9 z}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{9}{- 3} + \frac{- 9 z}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = \frac{9}{- 3} + \frac{- 9 z}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = \frac{9}{- 3} + \frac{- 9 z}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

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 $9$ by $- 3$ .

$y = - 3 + \frac{- 9 z}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.3.1.2

Cancel the common factor of $- 9$ and $- 3$ .

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Step 3.2.3.1.2.1

Factor $- 3$ out of $- 9 z$ .

$y = - 3 + \frac{- 3 (3 z)}{- 3}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $- 3$ out of $- 3$ .

$y = - 3 + \frac{- 3 (3 z)}{- 3 \cdot 1}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$y = - 3 + \frac{- 3 (3 z)}{- 3 \cdot 1}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$y = - 3 + \frac{3 z}{1}$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 3.2.3.1.2.2.4

Divide $3 z$ by $1$ .

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

$y = - 3 + 3 z$

$- 24 - 13 y + 12 z = 12$

$x = - 8 - 4 y + 3 z$

Step 4

Replace all occurrences of $y$ with $- 3 + 3 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 24 - 13 y + 12 z = 12$ with $- 3 + 3 z$ .

$- 24 - 13 (- 3 + 3 z) + 12 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 24 - 13 (- 3 + 3 z) + 12 z$ .

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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.

$- 24 - 13 \cdot - 3 - 13 (3 z) + 12 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

Step 4.2.1.1.2

Multiply $- 13$ by $- 3$ .

$- 24 + 39 - 13 (3 z) + 12 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

Step 4.2.1.1.3

Multiply $3$ by $- 13$ .

$- 24 + 39 - 39 z + 12 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

$- 24 + 39 - 39 z + 12 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 24$ and $39$ .

$15 - 39 z + 12 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

Step 4.2.1.2.2

Add $- 39 z$ and $12 z$ .

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = - 8 - 4 y + 3 z$

Step 4.3

Replace all occurrences of $y$ in $x = - 8 - 4 y + 3 z$ with $- 3 + 3 z$ .

$x = - 8 - 4 (- 3 + 3 z) + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 8 - 4 (- 3 + 3 z) + 3 z$ .

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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.

$x = - 8 - 4 \cdot - 3 - 4 (3 z) + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

Step 4.4.1.1.2

Multiply $- 4$ by $- 3$ .

$x = - 8 + 12 - 4 (3 z) + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

Step 4.4.1.1.3

Multiply $3$ by $- 4$ .

$x = - 8 + 12 - 12 z + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = - 8 + 12 - 12 z + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $- 8$ and $12$ .

$x = 4 - 12 z + 3 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

Step 4.4.1.2.2

Add $- 12 z$ and $3 z$ .

$x = 4 - 9 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = 4 - 9 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = 4 - 9 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = 4 - 9 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

$x = 4 - 9 z$

$15 - 27 z = 12$

$y = - 3 + 3 z$

Step 5

Solve for $z$ in $15 - 27 z = 12$ .

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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 $15$ from both sides of the equation.

$- 27 z = 12 - 15$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.1.2

Subtract $15$ from $12$ .

$- 27 z = - 3$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$- 27 z = - 3$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2

Divide each term in $- 27 z = - 3$ by $- 27$ and simplify.

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Step 5.2.1

Divide each term in $- 27 z = - 3$ by $- 27$ .

$\frac{- 27 z}{- 27} = \frac{- 3}{- 27}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 27$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 27 z}{- 27} = \frac{- 3}{- 27}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 3}{- 27}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{- 3}{- 27}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{- 3}{- 27}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $- 3$ and $- 27$ .

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Step 5.2.3.1.1

Factor $- 3$ out of $- 3$ .

$z = \frac{- 3 \cdot 1}{- 27}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $- 3$ out of $- 27$ .

$z = \frac{- 3 \cdot 1}{- 3 \cdot 9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2.3.1.2.2

Cancel the common factor.

$z = \frac{- 3 \cdot 1}{- 3 \cdot 9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 5.2.3.1.2.3

Rewrite the expression.

$z = \frac{1}{9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{1}{9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{1}{9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{1}{9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{1}{9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

$z = \frac{1}{9}$

$x = 4 - 9 z$

$y = - 3 + 3 z$

Step 6

Replace all occurrences of $z$ with $\frac{1}{9}$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = 4 - 9 z$ with $\frac{1}{9}$ .

$x = 4 - 9 (\frac{1}{9})$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor of $9$ .

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Step 6.2.1.1.1

Factor $9$ out of $- 9$ .

$x = 4 + 9 (- 1) (\frac{1}{9})$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

Step 6.2.1.1.2

Cancel the common factor.

$x = 4 + 9 \cdot (- 1 (\frac{1}{9}))$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

Step 6.2.1.1.3

Rewrite the expression.

$x = 4 - 1$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

$x = 4 - 1$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

Step 6.2.1.2

Subtract $1$ from $4$ .

$x = 3$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

$x = 3$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

$x = 3$

$z = \frac{1}{9}$

$y = - 3 + 3 z$

Step 6.3

Replace all occurrences of $z$ in $y = - 3 + 3 z$ with $\frac{1}{9}$ .

$y = - 3 + 3 (\frac{1}{9})$

$x = 3$

$z = \frac{1}{9}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 3 + 3 (\frac{1}{9})$ .

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Step 6.4.1.1

Cancel the common factor of $3$ .

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Step 6.4.1.1.1

Factor $3$ out of $9$ .

$y = - 3 + 3 (\frac{1}{3 (3)})$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.1.2

Cancel the common factor.

$y = - 3 + 3 (\frac{1}{3 \cdot 3})$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.1.3

Rewrite the expression.

$y = - 3 + \frac{1}{3}$

$x = 3$

$z = \frac{1}{9}$

$y = - 3 + \frac{1}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.2

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

$y = - 3 \cdot \frac{3}{3} + \frac{1}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.3

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

$y = \frac{- 3 \cdot 3}{3} + \frac{1}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$y = \frac{- 3 \cdot 3 + 1}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $- 3$ by $3$ .

$y = \frac{- 9 + 1}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.5.2

Add $- 9$ and $1$ .

$y = \frac{- 8}{3}$

$x = 3$

$z = \frac{1}{9}$

$y = \frac{- 8}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 6.4.1.6

Move the negative in front of the fraction.

$y = - \frac{8}{3}$

$x = 3$

$z = \frac{1}{9}$

$y = - \frac{8}{3}$

$x = 3$

$z = \frac{1}{9}$

$y = - \frac{8}{3}$

$x = 3$

$z = \frac{1}{9}$

$y = - \frac{8}{3}$

$x = 3$

$z = \frac{1}{9}$

Step 7

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

$(3, - \frac{8}{3}, \frac{1}{9})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, - \frac{8}{3}, \frac{1}{9})$

Equation Form:

$x = 3, y = - \frac{8}{3}, z = \frac{1}{9}$