Finite Math Examples

$3 x - 6 y + 6 z = - 3$ , $- 6 + 12 y - 12 z = 6$ , $6 x - 12 y + 12 z = - 6$

Step 1

Solve for $x$ in $3 x - 6 y + 6 z = - 3$ .

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Step 1.1

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

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Step 1.1.1

Add $6 y$ to both sides of the equation.

$3 x + 6 z = - 3 + 6 y$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.1.2

Subtract $6 z$ from both sides of the equation.

$3 x = - 3 + 6 y - 6 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$3 x = - 3 + 6 y - 6 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2

Divide each term in $3 x = - 3 + 6 y - 6 z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = - 3 + 6 y - 6 z$ by $3$ .

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

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

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

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

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

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $- 3$ by $3$ .

$x = - 1 + \frac{6 y}{3} + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.2

Cancel the common factor of $6$ and $3$ .

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Step 1.2.3.1.2.1

Factor $3$ out of $6 y$ .

$x = - 1 + \frac{3 (2 y)}{3} + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $3$ .

$x = - 1 + \frac{3 (2 y)}{3 (1)} + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - 1 + \frac{3 (2 y)}{3 \cdot 1} + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = - 1 + \frac{2 y}{1} + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.2.2.4

Divide $2 y$ by $1$ .

$x = - 1 + 2 y + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y + \frac{- 6 z}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.3

Cancel the common factor of $- 6$ and $3$ .

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Step 1.2.3.1.3.1

Factor $3$ out of $- 6 z$ .

$x = - 1 + 2 y + \frac{3 (- 2 z)}{3}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $3$ out of $3$ .

$x = - 1 + 2 y + \frac{3 (- 2 z)}{3 (1)}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.3.2.2

Cancel the common factor.

$x = - 1 + 2 y + \frac{3 (- 2 z)}{3 \cdot 1}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.3.2.3

Rewrite the expression.

$x = - 1 + 2 y + \frac{- 2 z}{1}$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 1.2.3.1.3.2.4

Divide $- 2 z$ by $1$ .

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$6 x - 12 y + 12 z = - 6$

Step 2

Replace all occurrences of $x$ with $- 1 + 2 y - 2 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $6 x - 12 y + 12 z = - 6$ with $- 1 + 2 y - 2 z$ .

$6 (- 1 + 2 y - 2 z) - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $6 (- 1 + 2 y - 2 z) - 12 y + 12 z$ .

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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.

$6 \cdot - 1 + 6 (2 y) + 6 (- 2 z) - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $6$ by $- 1$ .

$- 6 + 6 (2 y) + 6 (- 2 z) - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.1.2.2

Multiply $2$ by $6$ .

$- 6 + 12 y + 6 (- 2 z) - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.1.2.3

Multiply $- 2$ by $6$ .

$- 6 + 12 y - 12 z - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$- 6 + 12 y - 12 z - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$- 6 + 12 y - 12 z - 12 y + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.2

Combine the opposite terms in $- 6 + 12 y - 12 z - 12 y + 12 z$ .

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Step 2.2.1.2.1

Subtract $12 y$ from $12 y$ .

$- 6 - 12 z + 0 + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.2.2

Add $- 6 - 12 z$ and $0$ .

$- 6 - 12 z + 12 z = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.2.3

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

$- 6 + 0 = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 2.2.1.2.4

Add $- 6$ and $0$ .

$- 6 = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$- 6 = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$- 6 = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$- 6 = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

$- 6 = - 6$

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$

Step 3

Remove any equations from the system that are always true.

$x = - 1 + 2 y - 2 z$

$- 6 + 12 y - 12 z = 6$