Finite Math Examples

$6 x + 6 y + z = 60$ , $3 x + 4 y + z = 42$

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

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

$6 x + z = 60 - 6 y$

$3 x + 4 y + z = 42$

Step 1.1.2

Subtract $z$ from both sides of the equation.

$6 x = 60 - 6 y - z$

$3 x + 4 y + z = 42$

$6 x = 60 - 6 y - z$

$3 x + 4 y + z = 42$

Step 1.2

Divide each term in $6 x = 60 - 6 y - z$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = 60 - 6 y - z$ by $6$ .

$\frac{6 x}{6} = \frac{60}{6} + \frac{- 6 y}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{60}{6} + \frac{- 6 y}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{60}{6} + \frac{- 6 y}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

$x = \frac{60}{6} + \frac{- 6 y}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

$x = \frac{60}{6} + \frac{- 6 y}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

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

Divide $60$ by $6$ .

$x = 10 + \frac{- 6 y}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

Factor $6$ out of $- 6 y$ .

$x = 10 + \frac{6 (- y)}{6} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $6$ out of $6$ .

$x = 10 + \frac{6 (- y)}{6 (1)} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = 10 + \frac{6 (- y)}{6 \cdot 1} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 10 + \frac{- y}{1} + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.3.1.2.2.4

Divide $- y$ by $1$ .

$x = 10 - y + \frac{- z}{6}$

$3 x + 4 y + z = 42$

$x = 10 - y + \frac{- z}{6}$

$3 x + 4 y + z = 42$

$x = 10 - y + \frac{- z}{6}$

$3 x + 4 y + z = 42$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = 10 - y - \frac{z}{6}$

$3 x + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$3 x + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$3 x + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$3 x + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$3 x + 4 y + z = 42$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $3 (10 - y - \frac{z}{6}) + 4 y + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$3 \cdot 10 + 3 (- y) + 3 (- \frac{z}{6}) + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $3$ by $10$ .

$30 + 3 (- y) + 3 (- \frac{z}{6}) + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.1.2.2

Multiply $- 1$ by $3$ .

$30 - 3 y + 3 (- \frac{z}{6}) + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

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 $- \frac{z}{6}$ into the numerator.

$30 - 3 y + 3 (\frac{- z}{6}) + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.1.2.3.2

Factor $3$ out of $6$ .

$30 - 3 y + 3 (\frac{- z}{3 (2)}) + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.1.2.3.3

Cancel the common factor.

$30 - 3 y + 3 (\frac{- z}{3 \cdot 2}) + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.1.2.3.4

Rewrite the expression.

$30 - 3 y + \frac{- z}{2} + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$30 - 3 y + \frac{- z}{2} + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$30 - 3 y + \frac{- z}{2} + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.1.3

Move the negative in front of the fraction.

$30 - 3 y - \frac{z}{2} + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

$30 - 3 y - \frac{z}{2} + 4 y + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.2

Add $- 3 y$ and $4 y$ .

$30 + y - \frac{z}{2} + z = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.3

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

$30 + y - \frac{z}{2} + z \cdot \frac{2}{2} = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.4

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

$30 + y - \frac{z}{2} + \frac{z \cdot 2}{2} = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.5

Combine the numerators over the common denominator.

$30 + y + \frac{- z + z \cdot 2}{2} = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.6

Add $- z$ and $z \cdot 2$ .

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Step 2.1.6.1

Reorder $z$ and $2$ .

$30 + y + \frac{- z + 2 \cdot z}{2} = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.1.6.2

Add $- z$ and $2 \cdot z$ .

$30 + y + \frac{z}{2} = 42$

$x = 10 - y - \frac{z}{6}$

$30 + y + \frac{z}{2} = 42$

$x = 10 - y - \frac{z}{6}$

$30 + y + \frac{z}{2} = 42$

$x = 10 - y - \frac{z}{6}$

Step 2.2

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

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Step 2.2.1

Subtract $30$ from both sides of the equation.

$y + \frac{z}{2} = 42 - 30$

$x = 10 - y - \frac{z}{6}$

Step 2.2.2

Subtract $\frac{z}{2}$ from both sides of the equation.

$y = 42 - 30 - \frac{z}{2}$

$x = 10 - y - \frac{z}{6}$

Step 2.2.3

Subtract $30$ from $42$ .

$y = 12 - \frac{z}{2}$

$x = 10 - y - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

$x = 10 - y - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

$x = 10 - y - \frac{z}{6}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $10 - (12 - \frac{z}{2}) - \frac{z}{6}$ .

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Step 3.1.1

Find the common denominator.

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Step 3.1.1.1

Write $10$ as a fraction with denominator $1$ .

$x = \frac{10}{1} - (12 - \frac{z}{2}) - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.1.2

Multiply $\frac{10}{1}$ by $\frac{6}{6}$ .

$x = \frac{10}{1} \cdot \frac{6}{6} - (12 - \frac{z}{2}) - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.1.3

Multiply $\frac{10}{1}$ by $\frac{6}{6}$ .

$x = \frac{10 \cdot 6}{6} - (12 - \frac{z}{2}) - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.1.4

Write $- (12 - \frac{z}{2})$ as a fraction with denominator $1$ .

$x = \frac{10 \cdot 6}{6} + \frac{- (12 - \frac{z}{2})}{1} - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.1.5

Multiply $\frac{- (12 - \frac{z}{2})}{1}$ by $\frac{6}{6}$ .

$x = \frac{10 \cdot 6}{6} + \frac{- (12 - \frac{z}{2})}{1} \cdot \frac{6}{6} - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.1.6

Multiply $\frac{- (12 - \frac{z}{2})}{1}$ by $\frac{6}{6}$ .

$x = \frac{10 \cdot 6}{6} + \frac{- (12 - \frac{z}{2}) \cdot 6}{6} - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

$x = \frac{10 \cdot 6}{6} + \frac{- (12 - \frac{z}{2}) \cdot 6}{6} - \frac{z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.2

Combine the numerators over the common denominator.

$x = \frac{10 \cdot 6 - (12 - \frac{z}{2}) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Multiply $10$ by $6$ .

$x = \frac{60 - (12 - \frac{z}{2}) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.2

Apply the distributive property.

$x = \frac{60 + (- 1 \cdot 12 + \frac{z}{2}) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.3

Multiply $- 1$ by $12$ .

$x = \frac{60 + (- 12 + \frac{z}{2}) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.4

Multiply $- - \frac{z}{2}$ .

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Step 3.1.3.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{60 + (- 12 + 1 (\frac{z}{2})) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.4.2

Multiply $\frac{z}{2}$ by $1$ .

$x = \frac{60 + (- 12 + \frac{z}{2}) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

$x = \frac{60 + (- 12 + \frac{z}{2}) \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.5

Apply the distributive property.

$x = \frac{60 - 12 \cdot 6 + \frac{z}{2} \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.6

Multiply $- 12$ by $6$ .

$x = \frac{60 - 72 + \frac{z}{2} \cdot 6 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.7

Cancel the common factor of $2$ .

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Step 3.1.3.7.1

Factor $2$ out of $6$ .

$x = \frac{60 - 72 + \frac{z}{2} \cdot (2 (3)) - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.7.2

Cancel the common factor.

$x = \frac{60 - 72 + \frac{z}{2} \cdot (2 \cdot 3) - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.7.3

Rewrite the expression.

$x = \frac{60 - 72 + z \cdot 3 - z}{6}$

$y = 12 - \frac{z}{2}$

$x = \frac{60 - 72 + z \cdot 3 - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.3.8

Move $3$ to the left of $z$ .

$x = \frac{60 - 72 + 3 z - z}{6}$

$y = 12 - \frac{z}{2}$

$x = \frac{60 - 72 + 3 z - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

Subtract $72$ from $60$ .

$x = \frac{- 12 + 3 z - z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.2

Subtract $z$ from $3 z$ .

$x = \frac{- 12 + 2 z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.3

Cancel the common factor of $- 12 + 2 z$ and $6$ .

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Step 3.1.4.3.1

Factor $2$ out of $- 12$ .

$x = \frac{2 \cdot - 6 + 2 z}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.3.2

Factor $2$ out of $2 z$ .

$x = \frac{2 \cdot - 6 + 2 (z)}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.3.3

Factor $2$ out of $2 \cdot - 6 + 2 (z)$ .

$x = \frac{2 \cdot (- 6 + z)}{6}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.3.4

Cancel the common factors.

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Step 3.1.4.3.4.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot (- 6 + z)}{2 (3)}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.3.4.2

Cancel the common factor.

$x = \frac{2 \cdot (- 6 + z)}{2 \cdot 3}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.3.4.3

Rewrite the expression.

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

$y = 12 - \frac{z}{2}$

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

$y = 12 - \frac{z}{2}$

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

$y = 12 - \frac{z}{2}$

Step 3.1.4.4

Rewrite $- 6$ as $- 1 (6)$ .

$x = \frac{- 1 \cdot 6 + z}{3}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.5

Factor $- 1$ out of $z$ .

$x = \frac{- 1 \cdot 6 - 1 (- z)}{3}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.6

Factor $- 1$ out of $- 1 (6) - 1 (- z)$ .

$x = \frac{- 1 (6 - z)}{3}$

$y = 12 - \frac{z}{2}$

Step 3.1.4.7

Move the negative in front of the fraction.

$x = - \frac{6 - z}{3}$

$y = 12 - \frac{z}{2}$

$x = - \frac{6 - z}{3}$

$y = 12 - \frac{z}{2}$

$x = - \frac{6 - z}{3}$

$y = 12 - \frac{z}{2}$

$x = - \frac{6 - z}{3}$

$y = 12 - \frac{z}{2}$

Step 4

Simplify the right side.

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Step 4.1

Reorder $12$ and $- \frac{z}{2}$ .

$y = - \frac{z}{2} + 12$

$x = - \frac{6 - z}{3}$

$y = - \frac{z}{2} + 12$

$x = - \frac{6 - z}{3}$