Finite Math Examples

$x - 2 y - z = 2$ , $2 x - y + z = 4$ , $- x + y - 2 z = - 4$

Step 1

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

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Step 1.1

Add $2 y$ to both sides of the equation.

$x - z = 2 + 2 y$

$2 x - y + z = 4$

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

Step 1.2

Add $z$ to both sides of the equation.

$x = 2 + 2 y + z$

$2 x - y + z = 4$

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

$x = 2 + 2 y + z$

$2 x - y + z = 4$

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

Step 2

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

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Step 2.1

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

$2 (2 + 2 y + z) - y + z = 4$

$x = 2 + 2 y + z$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (2 + 2 y + z) - y + z$ .

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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.

$2 \cdot 2 + 2 (2 y) + 2 z - y + z = 4$

$x = 2 + 2 y + z$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $2$ .

$4 + 2 (2 y) + 2 z - y + z = 4$

$x = 2 + 2 y + z$

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

Step 2.2.1.1.2.2

Multiply $2$ by $2$ .

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

$x = 2 + 2 y + z$

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

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

$x = 2 + 2 y + z$

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

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

$x = 2 + 2 y + z$

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

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $y$ from $4 y$ .

$4 + 3 y + 2 z + z = 4$

$x = 2 + 2 y + z$

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

Step 2.2.1.2.2

Add $2 z$ and $z$ .

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

Step 2.3

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

$- (2 + 2 y + z) + y - 2 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- (2 + 2 y + z) + y - 2 z$ .

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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.

$- 1 \cdot 2 - (2 y) - z + y - 2 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $2$ .

$- 2 - (2 y) - z + y - 2 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 2.4.1.1.2.2

Multiply $2$ by $- 1$ .

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 2 y$ and $y$ .

$- 2 - y - z - 2 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 2.4.1.2.2

Subtract $2 z$ from $- z$ .

$- 2 - y - 3 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$- 2 - y - 3 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$- 2 - y - 3 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$- 2 - y - 3 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$- 2 - y - 3 z = - 4$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3

Solve for $y$ in $- 2 - y - 3 z = - 4$ .

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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 $2$ to both sides of the equation.

$- y - 3 z = - 4 + 2$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.1.2

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

$- y = - 4 + 2 + 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.1.3

Add $- 4$ and $2$ .

$- y = - 2 + 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$- y = - 2 + 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.2

Divide each term in $- y = - 2 + 3 z$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y = - 2 + 3 z$ by $- 1$ .

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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.

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.2.2.2

Divide $y$ by $1$ .

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

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 $- 1$ .

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

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{3 z}{- 1}$ .

$y = 2 - 1 \cdot (3 z)$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.2.3.1.3

Rewrite $- 1 \cdot (3 z)$ as $- (3 z)$ .

$y = 2 - (3 z)$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 3.2.3.1.4

Multiply $3$ by $- 1$ .

$y = 2 - 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$y = 2 - 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$y = 2 - 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$y = 2 - 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

$y = 2 - 3 z$

$4 + 3 y + 3 z = 4$

$x = 2 + 2 y + z$

Step 4

Replace all occurrences of $y$ with $2 - 3 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $4 + 3 y + 3 z = 4$ with $2 - 3 z$ .

$4 + 3 (2 - 3 z) + 3 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 + 3 (2 - 3 z) + 3 z$ .

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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.

$4 + 3 \cdot 2 + 3 (- 3 z) + 3 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

Step 4.2.1.1.2

Multiply $3$ by $2$ .

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

$y = 2 - 3 z$

$x = 2 + 2 y + z$

Step 4.2.1.1.3

Multiply $- 3$ by $3$ .

$4 + 6 - 9 z + 3 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

$4 + 6 - 9 z + 3 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $4$ and $6$ .

$10 - 9 z + 3 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

Step 4.2.1.2.2

Add $- 9 z$ and $3 z$ .

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 2 + 2 y + z$

Step 4.3

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

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

$10 - 6 z = 4$

$y = 2 - 3 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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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.

$x = 2 + 2 \cdot 2 + 2 (- 3 z) + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

Step 4.4.1.1.2

Multiply $2$ by $2$ .

$x = 2 + 4 + 2 (- 3 z) + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

Step 4.4.1.1.3

Multiply $- 3$ by $2$ .

$x = 2 + 4 - 6 z + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 2 + 4 - 6 z + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $2$ and $4$ .

$x = 6 - 6 z + z$

$10 - 6 z = 4$

$y = 2 - 3 z$

Step 4.4.1.2.2

Add $- 6 z$ and $z$ .

$x = 6 - 5 z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 6 - 5 z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 6 - 5 z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 6 - 5 z$

$10 - 6 z = 4$

$y = 2 - 3 z$

$x = 6 - 5 z$

$10 - 6 z = 4$

$y = 2 - 3 z$

Step 5

Solve for $z$ in $10 - 6 z = 4$ .

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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 $10$ from both sides of the equation.

$- 6 z = 4 - 10$

$x = 6 - 5 z$

$y = 2 - 3 z$

Step 5.1.2

Subtract $10$ from $4$ .

$- 6 z = - 6$

$x = 6 - 5 z$

$y = 2 - 3 z$

$- 6 z = - 6$

$x = 6 - 5 z$

$y = 2 - 3 z$

Step 5.2

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

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Step 5.2.1

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

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

$x = 6 - 5 z$

$y = 2 - 3 z$

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.

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

$x = 6 - 5 z$

$y = 2 - 3 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

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

$x = 6 - 5 z$

$y = 2 - 3 z$

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

$x = 6 - 5 z$

$y = 2 - 3 z$

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

$x = 6 - 5 z$

$y = 2 - 3 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 6$ by $- 6$ .

$z = 1$

$x = 6 - 5 z$

$y = 2 - 3 z$

$z = 1$

$x = 6 - 5 z$

$y = 2 - 3 z$

$z = 1$

$x = 6 - 5 z$

$y = 2 - 3 z$

$z = 1$

$x = 6 - 5 z$

$y = 2 - 3 z$

Step 6

Replace all occurrences of $z$ with $1$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = 6 - 5 z$ with $1$ .

$x = 6 - 5 \cdot 1$

$z = 1$

$y = 2 - 3 z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $6 - 5 \cdot 1$ .

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Step 6.2.1.1

Multiply $- 5$ by $1$ .

$x = 6 - 5$

$z = 1$

$y = 2 - 3 z$

Step 6.2.1.2

Subtract $5$ from $6$ .

$x = 1$

$z = 1$

$y = 2 - 3 z$

$x = 1$

$z = 1$

$y = 2 - 3 z$

$x = 1$

$z = 1$

$y = 2 - 3 z$

Step 6.3

Replace all occurrences of $z$ in $y = 2 - 3 z$ with $1$ .

$y = 2 - 3 \cdot 1$

$x = 1$

$z = 1$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $2 - 3 \cdot 1$ .

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Step 6.4.1.1

Multiply $- 3$ by $1$ .

$y = 2 - 3$

$x = 1$

$z = 1$

Step 6.4.1.2

Subtract $3$ from $2$ .

$y = - 1$

$x = 1$

$z = 1$

$y = - 1$

$x = 1$

$z = 1$

$y = - 1$

$x = 1$

$z = 1$

$y = - 1$

$x = 1$

$z = 1$

Step 7

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

$(1, - 1, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, - 1, 1)$

Equation Form:

$x = 1, y = - 1, z = 1$