Finite Math Examples

$2 x - y + z = 9$ , $3 x - y + z = 3$

Step 1

Solve the equation for $x$ .

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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 $y$ to both sides of the equation.

$2 x + z = 9 + y$

$3 x - y + z = 3$

Step 1.1.2

Subtract $z$ from both sides of the equation.

$2 x = 9 + y - z$

$3 x - y + z = 3$

$2 x = 9 + y - z$

$3 x - y + z = 3$

Step 1.2

Divide each term in $2 x = 9 + y - z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 9 + y - z$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2} + \frac{y}{2} + \frac{- z}{2}$

$3 x - y + z = 3$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2} + \frac{y}{2} + \frac{- z}{2}$

$3 x - y + z = 3$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{2} + \frac{y}{2} + \frac{- z}{2}$

$3 x - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} + \frac{- z}{2}$

$3 x - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} + \frac{- z}{2}$

$3 x - y + z = 3$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$3 x - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$3 x - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$3 x - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$3 x - y + z = 3$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $3 (\frac{9}{2} + \frac{y}{2} - \frac{z}{2}) - y + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$3 (\frac{9}{2}) + 3 (\frac{y}{2}) + 3 (- \frac{z}{2}) - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $3 (\frac{9}{2})$ .

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Step 2.1.1.2.1.1

Combine $3$ and $\frac{9}{2}$ .

$\frac{3 \cdot 9}{2} + 3 (\frac{y}{2}) + 3 (- \frac{z}{2}) - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.1.2.1.2

Multiply $3$ by $9$ .

$\frac{27}{2} + 3 (\frac{y}{2}) + 3 (- \frac{z}{2}) - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$\frac{27}{2} + 3 (\frac{y}{2}) + 3 (- \frac{z}{2}) - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.1.2.2

Combine $3$ and $\frac{y}{2}$ .

$\frac{27}{2} + \frac{3 y}{2} + 3 (- \frac{z}{2}) - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.1.2.3

Multiply $3 (- \frac{z}{2})$ .

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Step 2.1.1.2.3.1

Multiply $- 1$ by $3$ .

$\frac{27}{2} + \frac{3 y}{2} - 3 \frac{z}{2} - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.1.2.3.2

Combine $- 3$ and $\frac{z}{2}$ .

$\frac{27}{2} + \frac{3 y}{2} + \frac{- 3 z}{2} - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$\frac{27}{2} + \frac{3 y}{2} + \frac{- 3 z}{2} - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$\frac{27}{2} + \frac{3 y}{2} + \frac{- 3 z}{2} - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.1.3

Move the negative in front of the fraction.

$\frac{27}{2} + \frac{3 y}{2} - \frac{3 z}{2} - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$\frac{27}{2} + \frac{3 y}{2} - \frac{3 z}{2} - y + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.2

To write $- y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{27}{2} - \frac{3 z}{2} + \frac{3 y}{2} - y \cdot \frac{2}{2} + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.3

Combine $- y$ and $\frac{2}{2}$ .

$\frac{27}{2} - \frac{3 z}{2} + \frac{3 y}{2} + \frac{- y \cdot 2}{2} + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.4

Combine the numerators over the common denominator.

$\frac{27}{2} - \frac{3 z}{2} + \frac{3 y - y \cdot 2}{2} + z = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.5

Combine the numerators over the common denominator.

$z + \frac{27 - 3 z + 3 y - y \cdot 2}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.6

Multiply $2$ by $- 1$ .

$z + \frac{27 - 3 z + 3 y - 2 y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.7

Subtract $2 y$ from $3 y$ .

$z + \frac{27 - 3 z + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.8

To write $z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$z \cdot \frac{2}{2} + \frac{27 - 3 z + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.9

Simplify terms.

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Step 2.1.9.1

Combine $z$ and $\frac{2}{2}$ .

$\frac{z \cdot 2}{2} + \frac{27 - 3 z + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.9.2

Combine the numerators over the common denominator.

$\frac{z \cdot 2 + 27 - 3 z + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$\frac{z \cdot 2 + 27 - 3 z + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.10

Simplify the numerator.

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Step 2.1.10.1

Move $2$ to the left of $z$ .

$\frac{2 \cdot z + 27 - 3 z + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.10.2

Subtract $3 z$ from $2 z$ .

$\frac{- z + 27 + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$\frac{- z + 27 + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11

Simplify with factoring out.

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Step 2.1.11.1

Factor $- 1$ out of $- z$ .

$\frac{- (z) + 27 + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11.2

Rewrite $27$ as $- 1 (- 27)$ .

$\frac{- (z) - 1 \cdot - 27 + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11.3

Factor $- 1$ out of $- (z) - 1 (- 27)$ .

$\frac{- (z - 27) + y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11.4

Factor $- 1$ out of $y$ .

$\frac{- (z - 27) - 1 (- y)}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11.5

Factor $- 1$ out of $- (z - 27) - 1 (- y)$ .

$\frac{- (z - 27 - y)}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11.6

Simplify the expression.

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Step 2.1.11.6.1

Rewrite $- (z - 27 - y)$ as $- 1 (z - 27 - y)$ .

$\frac{- 1 (z - 27 - y)}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.1.11.6.2

Move the negative in front of the fraction.

$- \frac{z - 27 - y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$- \frac{z - 27 - y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$- \frac{z - 27 - y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$- \frac{z - 27 - y}{2} = 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.2

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{z - 27 - y}{2}) = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $- 2 (- \frac{z - 27 - y}{2})$ .

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Step 2.3.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1.1

Move the leading negative in $- \frac{z - 27 - y}{2}$ into the numerator.

$- 2 \frac{- (z - 27 - y)}{2} = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (z - 27 - y)}{2}) = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3.1.1.1.3

Cancel the common factor.

$2 \cdot (- 1 \frac{- (z - 27 - y)}{2}) = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3.1.1.1.4

Rewrite the expression.

$z - 27 - y = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$z - 27 - y = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3.1.1.2

Multiply.

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Step 2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (z - 27 - y) = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3.1.1.2.2

Multiply $z - 27 - y$ by $1$ .

$z - 27 - y = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$z - 27 - y = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$z - 27 - y = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$z - 27 - y = - 2 \cdot 3$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $- 2$ by $3$ .

$z - 27 - y = - 6$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$z - 27 - y = - 6$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$z - 27 - y = - 6$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.4

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

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Step 2.4.1

Subtract $z$ from both sides of the equation.

$- 27 - y = - 6 - z$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.4.2

Add $27$ to both sides of the equation.

$- y = - 6 - z + 27$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.4.3

Add $- 6$ and $27$ .

$- y = - z + 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$- y = - z + 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.5

Divide each term in $- y = - z + 21$ by $- 1$ and simplify.

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Step 2.5.1

Divide each term in $- y = - z + 21$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- z}{- 1} + \frac{21}{- 1}$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- z}{- 1} + \frac{21}{- 1}$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.5.2.2

Divide $y$ by $1$ .

$y = \frac{- z}{- 1} + \frac{21}{- 1}$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$y = \frac{- z}{- 1} + \frac{21}{- 1}$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{z}{1} + \frac{21}{- 1}$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.5.3.1.2

Divide $z$ by $1$ .

$y = z + \frac{21}{- 1}$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 2.5.3.1.3

Divide $21$ by $- 1$ .

$y = z - 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$y = z - 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$y = z - 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$y = z - 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

$y = z - 21$

$x = \frac{9}{2} + \frac{y}{2} - \frac{z}{2}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{9}{2} + \frac{z - 21}{2} - \frac{z}{2}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$x = \frac{9 + z - 21 - z}{2}$

$y = z - 21$

Step 3.1.2

Combine the opposite terms in $9 + z - 21 - z$ .

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Step 3.1.2.1

Subtract $z$ from $z$ .

$x = \frac{9 + 0 - 21}{2}$

$y = z - 21$

Step 3.1.2.2

Add $9$ and $0$ .

$x = \frac{9 - 21}{2}$

$y = z - 21$

$x = \frac{9 - 21}{2}$

$y = z - 21$

Step 3.1.3

Simplify the expression.

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Step 3.1.3.1

Subtract $21$ from $9$ .

$x = \frac{- 12}{2}$

$y = z - 21$

Step 3.1.3.2

Divide $- 12$ by $2$ .

$x = - 6$

$y = z - 21$

$x = - 6$

$y = z - 21$

$x = - 6$

$y = z - 21$

$x = - 6$

$y = z - 21$