Finite Math Examples

$x + y + z = - 12$ , $x - 2 y - z = 9$ , $2 x + 2 y - z = - 21$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = - 12 - y$

$x - 2 y - z = 9$

$2 x + 2 y - z = - 21$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = - 12 - y - z$

$x - 2 y - z = 9$

$2 x + 2 y - z = - 21$

$x = - 12 - y - z$

$x - 2 y - z = 9$

$2 x + 2 y - z = - 21$

Step 2

Replace all occurrences of $x$ with $- 12 - y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x - 2 y - z = 9$ with $- 12 - y - z$ .

$(- 12 - y - z) - 2 y - z = 9$

$x = - 12 - y - z$

$2 x + 2 y - z = - 21$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(- 12 - y - z) - 2 y - z$ .

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Step 2.2.1.1

Subtract $2 y$ from $- y$ .

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

$x = - 12 - y - z$

$2 x + 2 y - z = - 21$

Step 2.2.1.2

Subtract $z$ from $- z$ .

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$2 x + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$2 x + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$2 x + 2 y - z = - 21$

Step 2.3

Replace all occurrences of $x$ in $2 x + 2 y - z = - 21$ with $- 12 - y - z$ .

$2 (- 12 - y - z) + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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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.

$2 \cdot - 12 + 2 (- y) + 2 (- z) + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $- 12$ .

$- 24 + 2 (- y) + 2 (- z) + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $2$ .

$- 24 - 2 y + 2 (- z) + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $2$ .

$- 24 - 2 y - 2 z + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 2 y - 2 z + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 2 y - 2 z + 2 y - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Combine the opposite terms in $- 24 - 2 y - 2 z + 2 y - z$ .

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Step 2.4.1.2.1.1

Add $- 2 y$ and $2 y$ .

$- 24 - 2 z + 0 - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4.1.2.1.2

Add $- 24 - 2 z$ and $0$ .

$- 24 - 2 z - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 2 z - z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 2.4.1.2.2

Subtract $z$ from $- 2 z$ .

$- 24 - 3 z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 3 z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 3 z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 3 z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 24 - 3 z = - 21$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 3

Solve for $z$ in $- 24 - 3 z = - 21$ .

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Step 3.1

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

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Step 3.1.1

Add $24$ to both sides of the equation.

$- 3 z = - 21 + 24$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 3.1.2

Add $- 21$ and $24$ .

$- 3 z = 3$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$- 3 z = 3$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 3.2

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

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Step 3.2.1

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

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

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

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.

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

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 3.2.2.1.2

Divide $z$ by $1$ .

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

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

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

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

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

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $3$ by $- 3$ .

$z = - 1$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$z = - 1$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$z = - 1$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

$z = - 1$

$- 12 - 3 y - 2 z = 9$

$x = - 12 - y - z$

Step 4

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

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Step 4.1

Replace all occurrences of $z$ in $- 12 - 3 y - 2 z = 9$ with $- 1$ .

$- 12 - 3 y - 2 \cdot - 1 = 9$

$z = - 1$

$x = - 12 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 12 - 3 y - 2 \cdot - 1$ .

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Step 4.2.1.1

Multiply $- 2$ by $- 1$ .

$- 12 - 3 y + 2 = 9$

$z = - 1$

$x = - 12 - y - z$

Step 4.2.1.2

Add $- 12$ and $2$ .

$- 3 y - 10 = 9$

$z = - 1$

$x = - 12 - y - z$

$- 3 y - 10 = 9$

$z = - 1$

$x = - 12 - y - z$

$- 3 y - 10 = 9$

$z = - 1$

$x = - 12 - y - z$

Step 4.3

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

$x = - 12 - y - (- 1)$

$- 3 y - 10 = 9$

$z = - 1$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 12 - y - (- 1)$ .

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Step 4.4.1.1

Multiply $- 1$ by $- 1$ .

$x = - 12 - y + 1$

$- 3 y - 10 = 9$

$z = - 1$

Step 4.4.1.2

Add $- 12$ and $1$ .

$x = - y - 11$

$- 3 y - 10 = 9$

$z = - 1$

$x = - y - 11$

$- 3 y - 10 = 9$

$z = - 1$

$x = - y - 11$

$- 3 y - 10 = 9$

$z = - 1$

$x = - y - 11$

$- 3 y - 10 = 9$

$z = - 1$

Step 5

Solve for $y$ in $- 3 y - 10 = 9$ .

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Step 5.1

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

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Step 5.1.1

Add $10$ to both sides of the equation.

$- 3 y = 9 + 10$

$x = - y - 11$

$z = - 1$

Step 5.1.2

Add $9$ and $10$ .

$- 3 y = 19$

$x = - y - 11$

$z = - 1$

$- 3 y = 19$

$x = - y - 11$

$z = - 1$

Step 5.2

Divide each term in $- 3 y = 19$ by $- 3$ and simplify.

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Step 5.2.1

Divide each term in $- 3 y = 19$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{19}{- 3}$

$x = - y - 11$

$z = - 1$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{19}{- 3}$

$x = - y - 11$

$z = - 1$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{19}{- 3}$

$x = - y - 11$

$z = - 1$

$y = \frac{19}{- 3}$

$x = - y - 11$

$z = - 1$

$y = \frac{19}{- 3}$

$x = - y - 11$

$z = - 1$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$y = - \frac{19}{3}$

$x = - y - 11$

$z = - 1$

$y = - \frac{19}{3}$

$x = - y - 11$

$z = - 1$

$y = - \frac{19}{3}$

$x = - y - 11$

$z = - 1$

$y = - \frac{19}{3}$

$x = - y - 11$

$z = - 1$

Step 6

Replace all occurrences of $y$ with $- \frac{19}{3}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = - y - 11$ with $- \frac{19}{3}$ .

$x = - (- \frac{19}{3}) - 11$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- (- \frac{19}{3}) - 11$ .

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Step 6.2.1.1

Multiply $- (- \frac{19}{3})$ .

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Step 6.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{19}{3}) - 11$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.1.2

Multiply $\frac{19}{3}$ by $1$ .

$x = \frac{19}{3} - 11$

$y = - \frac{19}{3}$

$z = - 1$

$x = \frac{19}{3} - 11$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.2

To write $- 11$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{19}{3} - 11 \cdot \frac{3}{3}$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.3

Combine $- 11$ and $\frac{3}{3}$ .

$x = \frac{19}{3} + \frac{- 11 \cdot 3}{3}$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x = \frac{19 - 11 \cdot 3}{3}$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $- 11$ by $3$ .

$x = \frac{19 - 33}{3}$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.5.2

Subtract $33$ from $19$ .

$x = \frac{- 14}{3}$

$y = - \frac{19}{3}$

$z = - 1$

$x = \frac{- 14}{3}$

$y = - \frac{19}{3}$

$z = - 1$

Step 6.2.1.6

Move the negative in front of the fraction.

$x = - \frac{14}{3}$

$y = - \frac{19}{3}$

$z = - 1$

$x = - \frac{14}{3}$

$y = - \frac{19}{3}$

$z = - 1$

$x = - \frac{14}{3}$

$y = - \frac{19}{3}$

$z = - 1$

$x = - \frac{14}{3}$

$y = - \frac{19}{3}$

$z = - 1$

Step 7

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

$(- \frac{14}{3}, - \frac{19}{3}, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- \frac{14}{3}, - \frac{19}{3}, - 1)$

Equation Form:

$x = - \frac{14}{3}, y = - \frac{19}{3}, z = - 1$