Finite Math Examples

$z = 30 x + 40 y$

Step 1

Solve the equation for $y$ .

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Step 1.1

Rewrite the equation as $30 x + 40 y = z$ .

$30 x + 40 y = z$

Step 1.2

Subtract $30 x$ from both sides of the equation.

$40 y = z - 30 x$

Step 1.3

Divide each term in $40 y = z - 30 x$ by $40$ and simplify.

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Step 1.3.1

Divide each term in $40 y = z - 30 x$ by $40$ .

$\frac{40 y}{40} = \frac{z}{40} + \frac{- 30 x}{40}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $40$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{40 y}{40} = \frac{z}{40} + \frac{- 30 x}{40}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{z}{40} + \frac{- 30 x}{40}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Cancel the common factor of $- 30$ and $40$ .

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Step 1.3.3.1.1.1

Factor $10$ out of $- 30 x$ .

$y = \frac{z}{40} + \frac{10 (- 3 x)}{40}$

Step 1.3.3.1.1.2

Cancel the common factors.

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Step 1.3.3.1.1.2.1

Factor $10$ out of $40$ .

$y = \frac{z}{40} + \frac{10 (- 3 x)}{10 (4)}$

Step 1.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{z}{40} + \frac{10 (- 3 x)}{10 \cdot 4}$

Step 1.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{z}{40} + \frac{- 3 x}{4}$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$y = \frac{z}{40} - \frac{3 x}{4}$

Step 2

Choose any values for $x$ and $y$ that are in the domain to plug into the equation.

Step 3

Choose $0$ to substitute in for $x$ and $1$ to substitute for $z$ .

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Step 3.1

Remove parentheses.

$y = \frac{1}{40} - \frac{3 (0)}{4}$

Step 3.2

Simplify $\frac{1}{40} - \frac{3 (0)}{4}$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $0$ and $4$ .

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Step 3.2.1.1.1

Factor $4$ out of $3 (0)$ .

$y = \frac{1}{40} - \frac{4 (3 \cdot (0))}{4}$

Step 3.2.1.1.2

Cancel the common factors.

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Step 3.2.1.1.2.1

Factor $4$ out of $4$ .

$y = \frac{1}{40} - \frac{4 (3 \cdot (0))}{4 (1)}$

Step 3.2.1.1.2.2

Cancel the common factor.

$y = \frac{1}{40} - \frac{4 (3 \cdot (0))}{4 \cdot 1}$

Step 3.2.1.1.2.3

Rewrite the expression.

$y = \frac{1}{40} - \frac{3 \cdot (0)}{1}$

Step 3.2.1.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$y = \frac{1}{40} - (3 \cdot (0))$

Step 3.2.1.2

Multiply $- (3 \cdot (0))$ .

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Step 3.2.1.2.1

Multiply $3$ by $0$ .

$y = \frac{1}{40} - 0$

Step 3.2.1.2.2

Multiply $- 1$ by $0$ .

$y = \frac{1}{40} + 0$

Step 3.2.2

Add $\frac{1}{40}$ and $0$ .

$y = \frac{1}{40}$

Step 3.3

Use the $x$ , $y$ , and $z$ values to form the ordered pair.

$(0, \frac{1}{40}, 1)$

Step 4

Choose $1$ to substitute in for $x$ and $2$ to substitute for $z$ .

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Step 4.1

Remove parentheses.

$y = \frac{2}{40} - \frac{3 (1)}{4}$

Step 4.2

Simplify $\frac{2}{40} - \frac{3 (1)}{4}$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $2$ and $40$ .

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Step 4.2.1.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{40} - \frac{3 (1)}{4}$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $2$ out of $40$ .

$y = \frac{2 \cdot 1}{2 \cdot 20} - \frac{3 (1)}{4}$

Step 4.2.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot 20} - \frac{3 (1)}{4}$

Step 4.2.1.1.2.3

Rewrite the expression.

$y = \frac{1}{20} - \frac{3 (1)}{4}$

Step 4.2.1.2

Multiply $3$ by $1$ .

$y = \frac{1}{20} - \frac{3}{4}$

Step 4.2.2

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{1}{20} - \frac{3}{4} \cdot \frac{5}{5}$

Step 4.2.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{3}{4}$ by $\frac{5}{5}$ .

$y = \frac{1}{20} - \frac{3 \cdot 5}{4 \cdot 5}$

Step 4.2.3.2

Multiply $4$ by $5$ .

$y = \frac{1}{20} - \frac{3 \cdot 5}{20}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 3 \cdot 5}{20}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 3$ by $5$ .

$y = \frac{1 - 15}{20}$

Step 4.2.5.2

Subtract $15$ from $1$ .

$y = \frac{- 14}{20}$

Step 4.2.6

Cancel the common factor of $- 14$ and $20$ .

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Step 4.2.6.1

Factor $2$ out of $- 14$ .

$y = \frac{2 (- 7)}{20}$

Step 4.2.6.2

Cancel the common factors.

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Step 4.2.6.2.1

Factor $2$ out of $20$ .

$y = \frac{2 \cdot - 7}{2 \cdot 10}$

Step 4.2.6.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 7}{2 \cdot 10}$

Step 4.2.6.2.3

Rewrite the expression.

$y = \frac{- 7}{10}$

Step 4.2.7

Move the negative in front of the fraction.

$y = - \frac{7}{10}$

Step 4.3

Use the $x$ , $y$ , and $z$ values to form the ordered pair.

$(1, - \frac{7}{10}, 2)$

Step 5

Choose $2$ to substitute in for $x$ and $3$ to substitute for $z$ .

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Step 5.1

Remove parentheses.

$y = \frac{3}{40} - \frac{3 (2)}{4}$

Step 5.2

Simplify $\frac{3}{40} - \frac{3 (2)}{4}$ .

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Step 5.2.1

Cancel the common factor of $2$ and $4$ .

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Step 5.2.1.1

Factor $2$ out of $3 (2)$ .

$y = \frac{3}{40} - \frac{2 \cdot 3}{4}$

Step 5.2.1.2

Cancel the common factors.

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Step 5.2.1.2.1

Factor $2$ out of $4$ .

$y = \frac{3}{40} - \frac{2 \cdot 3}{2 \cdot 2}$

Step 5.2.1.2.2

Cancel the common factor.

$y = \frac{3}{40} - \frac{2 \cdot 3}{2 \cdot 2}$

Step 5.2.1.2.3

Rewrite the expression.

$y = \frac{3}{40} - \frac{3}{2}$

Step 5.2.2

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$y = \frac{3}{40} - \frac{3}{2} \cdot \frac{20}{20}$

Step 5.2.3

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.1

Multiply $\frac{3}{2}$ by $\frac{20}{20}$ .

$y = \frac{3}{40} - \frac{3 \cdot 20}{2 \cdot 20}$

Step 5.2.3.2

Multiply $2$ by $20$ .

$y = \frac{3}{40} - \frac{3 \cdot 20}{40}$

Step 5.2.4

Combine the numerators over the common denominator.

$y = \frac{3 - 3 \cdot 20}{40}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 3$ by $20$ .

$y = \frac{3 - 60}{40}$

Step 5.2.5.2

Subtract $60$ from $3$ .

$y = \frac{- 57}{40}$

Step 5.2.6

Move the negative in front of the fraction.

$y = - \frac{57}{40}$

Step 5.3

Use the $x$ , $y$ , and $z$ values to form the ordered pair.

$(2, - \frac{57}{40}, 3)$

Step 6

These are three possible solutions to the equation.

$(0, \frac{1}{40}, 1), (1, - \frac{7}{10}, 2), (2, - \frac{57}{40}, 3)$