Finite Math Examples

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

Step 1

Reorder $1$ and $- y$ .

$x = - y + 1$

$2 x = z$

$2 z = - 2 - y$

Step 2

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

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Step 2.1

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

$2 (- y + 1) = z$

$x = - y + 1$

$2 z = - 2 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- y + 1)$ .

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Step 2.2.1.1

Apply the distributive property.

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

$x = - y + 1$

$2 z = - 2 - y$

Step 2.2.1.2

Multiply.

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Step 2.2.1.2.1

Multiply $- 1$ by $2$ .

$- 2 y + 2 \cdot 1 = z$

$x = - y + 1$

$2 z = - 2 - y$

Step 2.2.1.2.2

Multiply $2$ by $1$ .

$- 2 y + 2 = z$

$x = - y + 1$

$2 z = - 2 - y$

$- 2 y + 2 = z$

$x = - y + 1$

$2 z = - 2 - y$

$- 2 y + 2 = z$

$x = - y + 1$

$2 z = - 2 - y$

$- 2 y + 2 = z$

$x = - y + 1$

$2 z = - 2 - y$

$- 2 y + 2 = z$

$x = - y + 1$

$2 z = - 2 - y$

Step 3

Rewrite the equation as $z = - 2 y + 2$ .

$z = - 2 y + 2$

$x = - y + 1$

$2 z = - 2 - y$

Step 4

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

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Step 4.1

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

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

$z = - 2 y + 2$

$x = - y + 1$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Apply the distributive property.

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

$z = - 2 y + 2$

$x = - y + 1$

Step 4.2.1.2

Multiply.

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Step 4.2.1.2.1

Multiply $- 2$ by $2$ .

$- 4 y + 2 \cdot 2 = - 2 - y$

$z = - 2 y + 2$

$x = - y + 1$

Step 4.2.1.2.2

Multiply $2$ by $2$ .

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

$z = - 2 y + 2$

$x = - y + 1$

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

$z = - 2 y + 2$

$x = - y + 1$

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

$z = - 2 y + 2$

$x = - y + 1$

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

$z = - 2 y + 2$

$x = - y + 1$

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

$z = - 2 y + 2$

$x = - y + 1$

Step 5

Solve for $y$ in $- 4 y + 4 = - 2 - y$ .

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Step 5.1

Move all terms containing $y$ to the left side of the equation.

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Step 5.1.1

Add $y$ to both sides of the equation.

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

$z = - 2 y + 2$

$x = - y + 1$

Step 5.1.2

Add $- 4 y$ and $y$ .

$- 3 y + 4 = - 2$

$z = - 2 y + 2$

$x = - y + 1$

$- 3 y + 4 = - 2$

$z = - 2 y + 2$

$x = - y + 1$

Step 5.2

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

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Step 5.2.1

Subtract $4$ from both sides of the equation.

$- 3 y = - 2 - 4$

$z = - 2 y + 2$

$x = - y + 1$

Step 5.2.2

Subtract $4$ from $- 2$ .

$- 3 y = - 6$

$z = - 2 y + 2$

$x = - y + 1$

$- 3 y = - 6$

$z = - 2 y + 2$

$x = - y + 1$

Step 5.3

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

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Step 5.3.1

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

$\frac{- 3 y}{- 3} = \frac{- 6}{- 3}$

$z = - 2 y + 2$

$x = - y + 1$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- 6}{- 3}$

$z = - 2 y + 2$

$x = - y + 1$

Step 5.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 6}{- 3}$

$z = - 2 y + 2$

$x = - y + 1$

$y = \frac{- 6}{- 3}$

$z = - 2 y + 2$

$x = - y + 1$

$y = \frac{- 6}{- 3}$

$z = - 2 y + 2$

$x = - y + 1$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 6$ by $- 3$ .

$y = 2$

$z = - 2 y + 2$

$x = - y + 1$

$y = 2$

$z = - 2 y + 2$

$x = - y + 1$

$y = 2$

$z = - 2 y + 2$

$x = - y + 1$

$y = 2$

$z = - 2 y + 2$

$x = - y + 1$

Step 6

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

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Step 6.1

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

$z = - 2 \cdot 2 + 2$

$y = 2$

$x = - y + 1$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 2 \cdot 2 + 2$ .

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Step 6.2.1.1

Multiply $- 2$ by $2$ .

$z = - 4 + 2$

$y = 2$

$x = - y + 1$

Step 6.2.1.2

Add $- 4$ and $2$ .

$z = - 2$

$y = 2$

$x = - y + 1$

$z = - 2$

$y = 2$

$x = - y + 1$

$z = - 2$

$y = 2$

$x = - y + 1$

Step 6.3

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

$x = - (2) + 1$

$z = - 2$

$y = 2$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- (2) + 1$ .

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Step 6.4.1.1

Multiply $- 1$ by $2$ .

$x = - 2 + 1$

$z = - 2$

$y = 2$

Step 6.4.1.2

Add $- 2$ and $1$ .

$x = - 1$

$z = - 2$

$y = 2$

$x = - 1$

$z = - 2$

$y = 2$

$x = - 1$

$z = - 2$

$y = 2$

$x = - 1$

$z = - 2$

$y = 2$

Step 7

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

$(- 1, 2, - 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1, 2, - 2)$

Equation Form:

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