Finite Math Examples

$x + 3 y + 6 z = 20$ , $2 x - 3 y + 3 z = 4$

Step 1

Solve the equation for $y$ .

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Step 1.1

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

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$3 y + 6 z = 20 - x$

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

Step 1.1.2

Subtract $6 z$ from both sides of the equation.

$3 y = 20 - x - 6 z$

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

$3 y = 20 - x - 6 z$

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

Step 1.2

Divide each term in $3 y = 20 - x - 6 z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 y = 20 - x - 6 z$ by $3$ .

$\frac{3 y}{3} = \frac{20}{3} + \frac{- x}{3} + \frac{- 6 z}{3}$

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{20}{3} + \frac{- x}{3} + \frac{- 6 z}{3}$

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{20}{3} + \frac{- x}{3} + \frac{- 6 z}{3}$

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

$y = \frac{20}{3} + \frac{- x}{3} + \frac{- 6 z}{3}$

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

$y = \frac{20}{3} + \frac{- x}{3} + \frac{- 6 z}{3}$

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = \frac{20}{3} - \frac{x}{3} + \frac{- 6 z}{3}$

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

Step 1.2.3.1.2

Cancel the common factor of $- 6$ and $3$ .

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Step 1.2.3.1.2.1

Factor $3$ out of $- 6 z$ .

$y = \frac{20}{3} - \frac{x}{3} + \frac{3 (- 2 z)}{3}$

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

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $3$ .

$y = \frac{20}{3} - \frac{x}{3} + \frac{3 (- 2 z)}{3 (1)}$

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

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{20}{3} - \frac{x}{3} + \frac{3 (- 2 z)}{3 \cdot 1}$

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

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{20}{3} - \frac{x}{3} + \frac{- 2 z}{1}$

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

Step 1.2.3.1.2.2.4

Divide $- 2 z$ by $1$ .

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

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

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $2 x - 3 (\frac{20}{3} - \frac{x}{3} - 2 z) + 3 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$2 x - 3 (\frac{20}{3}) - 3 (- \frac{x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Cancel the common factor of $3$ .

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Step 2.1.1.2.1.1

Factor $3$ out of $- 3$ .

$2 x + 3 (- 1) (\frac{20}{3}) - 3 (- \frac{x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.1.2

Cancel the common factor.

$2 x + 3 \cdot (- 1 (\frac{20}{3})) - 3 (- \frac{x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.1.3

Rewrite the expression.

$2 x - 1 \cdot 20 - 3 (- \frac{x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$2 x - 1 \cdot 20 - 3 (- \frac{x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.2

Multiply $- 1$ by $20$ .

$2 x - 20 - 3 (- \frac{x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.3

Cancel the common factor of $3$ .

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Step 2.1.1.2.3.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$2 x - 20 - 3 \frac{- x}{3} - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.3.2

Factor $3$ out of $- 3$ .

$2 x - 20 + 3 (- 1) (\frac{- x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.3.3

Cancel the common factor.

$2 x - 20 + 3 \cdot (- 1 \frac{- x}{3}) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.3.4

Rewrite the expression.

$2 x - 20 - 1 (- x) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$2 x - 20 - 1 (- x) - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.4

Multiply $- 1$ by $- 1$ .

$2 x - 20 + 1 x - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.5

Multiply $x$ by $1$ .

$2 x - 20 + x - 3 (- 2 z) + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.1.2.6

Multiply $- 2$ by $- 3$ .

$2 x - 20 + x + 6 z + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$2 x - 20 + x + 6 z + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$2 x - 20 + x + 6 z + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Add $2 x$ and $x$ .

$3 x - 20 + 6 z + 3 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.1.2.2

Add $6 z$ and $3 z$ .

$3 x - 20 + 9 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$3 x - 20 + 9 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$3 x - 20 + 9 z = 4$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.2

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

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Step 2.2.1

Subtract $3 x$ from both sides of the equation.

$- 20 + 9 z = 4 - 3 x$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.2.2

Add $20$ to both sides of the equation.

$9 z = 4 - 3 x + 20$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.2.3

Add $4$ and $20$ .

$9 z = - 3 x + 24$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$9 z = - 3 x + 24$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3

Divide each term in $9 z = - 3 x + 24$ by $9$ and simplify.

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Step 2.3.1

Divide each term in $9 z = - 3 x + 24$ by $9$ .

$\frac{9 z}{9} = \frac{- 3 x}{9} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $9$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{9 z}{9} = \frac{- 3 x}{9} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 3 x}{9} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = \frac{- 3 x}{9} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = \frac{- 3 x}{9} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Cancel the common factor of $- 3$ and $9$ .

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Step 2.3.3.1.1.1

Factor $3$ out of $- 3 x$ .

$z = \frac{3 (- x)}{9} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

Factor $3$ out of $9$ .

$z = \frac{3 (- x)}{3 (3)} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.1.2.2

Cancel the common factor.

$z = \frac{3 (- x)}{3 \cdot 3} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.1.2.3

Rewrite the expression.

$z = \frac{- x}{3} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = \frac{- x}{3} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = \frac{- x}{3} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.2

Move the negative in front of the fraction.

$z = - \frac{x}{3} + \frac{24}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.3

Cancel the common factor of $24$ and $9$ .

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Step 2.3.3.1.3.1

Factor $3$ out of $24$ .

$z = - \frac{x}{3} + \frac{3 (8)}{9}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.3.2

Cancel the common factors.

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Step 2.3.3.1.3.2.1

Factor $3$ out of $9$ .

$z = - \frac{x}{3} + \frac{3 \cdot 8}{3 \cdot 3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.3.2.2

Cancel the common factor.

$z = - \frac{x}{3} + \frac{3 \cdot 8}{3 \cdot 3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 2.3.3.1.3.2.3

Rewrite the expression.

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} - 2 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{20}{3} - \frac{x}{3} - 2 (- \frac{x}{3} + \frac{8}{3})$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$y = \frac{20}{3} - \frac{x}{3} - 2 (- \frac{x}{3}) - 2 (\frac{8}{3})$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.1.2

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

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Step 3.1.1.2.1

Multiply $- 1$ by $- 2$ .

$y = \frac{20}{3} - \frac{x}{3} + 2 (\frac{x}{3}) - 2 (\frac{8}{3})$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.1.2.2

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

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} - 2 (\frac{8}{3})$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} - 2 (\frac{8}{3})$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.1.3

Multiply $- 2 (\frac{8}{3})$ .

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Step 3.1.1.3.1

Combine $- 2$ and $\frac{8}{3}$ .

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} + \frac{- 2 \cdot 8}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.1.3.2

Multiply $- 2$ by $8$ .

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} + \frac{- 16}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} + \frac{- 16}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.1.4

Move the negative in front of the fraction.

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} - \frac{16}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{20}{3} - \frac{x}{3} + \frac{2 x}{3} - \frac{16}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.2

Simplify terms.

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Step 3.1.2.1

Combine the numerators over the common denominator.

$y = \frac{20 - x + 2 x - 16}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.2.2

Subtract $16$ from $20$ .

$y = \frac{- x + 2 x + 4}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

Step 3.1.2.3

Add $- x$ and $2 x$ .

$y = \frac{x + 4}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{x + 4}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{x + 4}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$

$y = \frac{x + 4}{3}$

$z = - \frac{x}{3} + \frac{8}{3}$