Finite Math Examples

$- x + y + z = - 2$ , $- x + 5 y - 11 z = - 14$ , $3 x - y - 9 z = 0$

Step 1

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

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Step 1.1

Add $x$ to both sides of the equation.

$y + z = - 2 + x$

$- x + 5 y - 11 z = - 14$

$3 x - y - 9 z = 0$

Step 1.2

Subtract $z$ from both sides of the equation.

$y = - 2 + x - z$

$- x + 5 y - 11 z = - 14$

$3 x - y - 9 z = 0$

$y = - 2 + x - z$

$- x + 5 y - 11 z = - 14$

$3 x - y - 9 z = 0$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $- x + 5 y - 11 z = - 14$ with $- 2 + x - z$ .

$- x + 5 (- 2 + x - z) - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- x + 5 (- 2 + x - z) - 11 z$ .

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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.

$- x + 5 \cdot - 2 + 5 x + 5 (- z) - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $5$ by $- 2$ .

$- x - 10 + 5 x + 5 (- z) - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

Step 2.2.1.1.2.2

Multiply $- 1$ by $5$ .

$- x - 10 + 5 x - 5 z - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

$- x - 10 + 5 x - 5 z - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

$- x - 10 + 5 x - 5 z - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- x$ and $5 x$ .

$4 x - 10 - 5 z - 11 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

Step 2.2.1.2.2

Subtract $11 z$ from $- 5 z$ .

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x - y - 9 z = 0$

Step 2.3

Replace all occurrences of $y$ in $3 x - y - 9 z = 0$ with $- 2 + x - z$ .

$3 x - (- 2 + x - z) - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 x - (- 2 + x - z) - 9 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$3 x + 2 - x + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 2$ .

$3 x + 2 - x + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 2.4.1.1.2.2

Multiply $- - z$ .

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Step 2.4.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$3 x + 2 - x + 1 z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 2.4.1.1.2.2.2

Multiply $z$ by $1$ .

$3 x + 2 - x + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x + 2 - x + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x + 2 - x + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$3 x + 2 - x + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $x$ from $3 x$ .

$2 x + 2 + z - 9 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 2.4.1.2.2

Subtract $9 z$ from $z$ .

$2 x + 2 - 8 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$2 x + 2 - 8 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$2 x + 2 - 8 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$2 x + 2 - 8 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$2 x + 2 - 8 z = 0$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3

Solve for $x$ in $2 x + 2 - 8 z = 0$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2$ from both sides of the equation.

$2 x - 8 z = - 2$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.1.2

Add $8 z$ to both sides of the equation.

$2 x = - 2 + 8 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$2 x = - 2 + 8 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2

Divide each term in $2 x = - 2 + 8 z$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = - 2 + 8 z$ by $2$ .

$\frac{2 x}{2} = \frac{- 2}{2} + \frac{8 z}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 2}{2} + \frac{8 z}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{2} + \frac{8 z}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = \frac{- 2}{2} + \frac{8 z}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = \frac{- 2}{2} + \frac{8 z}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

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 $- 2$ by $2$ .

$x = - 1 + \frac{8 z}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.3.1.2

Cancel the common factor of $8$ and $2$ .

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Step 3.2.3.1.2.1

Factor $2$ out of $8 z$ .

$x = - 1 + \frac{2 (4 z)}{2}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $2$ out of $2$ .

$x = - 1 + \frac{2 (4 z)}{2 (1)}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$x = - 1 + \frac{2 (4 z)}{2 \cdot 1}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$x = - 1 + \frac{4 z}{1}$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 3.2.3.1.2.2.4

Divide $4 z$ by $1$ .

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

$x = - 1 + 4 z$

$4 x - 10 - 16 z = - 14$

$y = - 2 + x - z$

Step 4

Replace all occurrences of $x$ with $- 1 + 4 z$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $4 x - 10 - 16 z = - 14$ with $- 1 + 4 z$ .

$4 (- 1 + 4 z) - 10 - 16 z = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 (- 1 + 4 z) - 10 - 16 z$ .

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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.

$4 \cdot - 1 + 4 (4 z) - 10 - 16 z = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.2.1.1.2

Multiply $4$ by $- 1$ .

$- 4 + 4 (4 z) - 10 - 16 z = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.2.1.1.3

Multiply $4$ by $4$ .

$- 4 + 16 z - 10 - 16 z = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

$- 4 + 16 z - 10 - 16 z = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $- 4 + 16 z - 10 - 16 z$ .

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Step 4.2.1.2.1.1

Subtract $16 z$ from $16 z$ .

$- 4 + 0 - 10 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.2.1.2.1.2

Add $- 4$ and $0$ .

$- 4 - 10 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

$- 4 - 10 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.2.1.2.2

Subtract $10$ from $- 4$ .

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 2 + x - z$

Step 4.3

Replace all occurrences of $x$ in $y = - 2 + x - z$ with $- 1 + 4 z$ .

$y = - 2 - 1 + 4 z - z$

$- 14 = - 14$

$x = - 1 + 4 z$

Step 4.4

Simplify $y = - 2 - 1 + 4 z - z$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$y = - 2 - 1 + 4 z - z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 2 - 1 + 4 z - z$

$- 14 = - 14$

$x = - 1 + 4 z$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- 2 - 1 + 4 z - z$ .

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Step 4.4.2.1.1

Subtract $1$ from $- 2$ .

$y = - 3 + 4 z - z$

$- 14 = - 14$

$x = - 1 + 4 z$

Step 4.4.2.1.2

Subtract $z$ from $4 z$ .

$y = - 3 + 3 z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 3 + 3 z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 3 + 3 z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 3 + 3 z$

$- 14 = - 14$

$x = - 1 + 4 z$

$y = - 3 + 3 z$

$- 14 = - 14$

$x = - 1 + 4 z$

Step 5

Remove any equations from the system that are always true.

$y = - 3 + 3 z$

$x = - 1 + 4 z$