Finite Math Examples

$0.5 x + 0.1 y = 0.9$ , $0.1 x - 0.1 y = - 0.1$ , $x + y = \frac{11}{3}$

Step 1

Subtract $y$ from both sides of the equation.

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

$0.5 x + 0.1 y = 0.9$

$0.1 x - 0.1 y = - 0.1$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $0.5 x + 0.1 y = 0.9$ with $\frac{11}{3} - y$ .

$0.5 (\frac{11}{3} - y) + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $0.5 (\frac{11}{3} - y) + 0.1 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$0.5 (\frac{11}{3}) + 0.5 (- y) + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.2.1.1.2

Cancel the common factor of $0.5$ .

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Step 2.2.1.1.2.1

Factor $0.5$ out of $3$ .

$0.5 (\frac{11}{0.5 (6)}) + 0.5 (- y) + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.2.1.1.2.2

Cancel the common factor.

$0.5 (\frac{11}{0.5 \cdot 6}) + 0.5 (- y) + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.2.1.1.2.3

Rewrite the expression.

$\frac{11}{6} + 0.5 (- y) + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

$\frac{11}{6} + 0.5 (- y) + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.2.1.1.3

Multiply $- 1$ by $0.5$ .

$\frac{11}{6} - 0.5 y + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

$\frac{11}{6} - 0.5 y + 0.1 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.2.1.2

Add $- 0.5 y$ and $0.1 y$ .

$\frac{11}{6} - 0.4 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$0.1 x - 0.1 y = - 0.1$

Step 2.3

Replace all occurrences of $x$ in $0.1 x - 0.1 y = - 0.1$ with $\frac{11}{3} - y$ .

$0.1 (\frac{11}{3} - y) - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $0.1 (\frac{11}{3} - y) - 0.1 y$ .

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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.

$0.1 (\frac{11}{3}) + 0.1 (- y) - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 2.4.1.1.2

Cancel the common factor of $0.1$ .

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Step 2.4.1.1.2.1

Factor $0.1$ out of $3$ .

$0.1 (\frac{11}{0.1 (30)}) + 0.1 (- y) - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 2.4.1.1.2.2

Cancel the common factor.

$0.1 (\frac{11}{0.1 \cdot 30}) + 0.1 (- y) - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 2.4.1.1.2.3

Rewrite the expression.

$\frac{11}{30} + 0.1 (- y) - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$\frac{11}{30} + 0.1 (- y) - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 2.4.1.1.3

Multiply $- 1$ by $0.1$ .

$\frac{11}{30} - 0.1 y - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$\frac{11}{30} - 0.1 y - 0.1 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 2.4.1.2

Subtract $0.1 y$ from $- 0.1 y$ .

$\frac{11}{30} - 0.2 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$\frac{11}{30} - 0.2 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$\frac{11}{30} - 0.2 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

$\frac{11}{30} - 0.2 y = - 0.1$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3

Solve for $y$ in $\frac{11}{30} - 0.2 y = - 0.1$ .

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Step 3.1

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

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Step 3.1.1

Subtract $\frac{11}{30}$ from both sides of the equation.

$- 0.2 y = - 0.1 - \frac{11}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.2

To write $- 0.1$ as a fraction with a common denominator, multiply by $\frac{30}{30}$ .

$- 0.2 y = - 0.1 \cdot \frac{30}{30} - \frac{11}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.3

Combine $- 0.1$ and $\frac{30}{30}$ .

$- 0.2 y = \frac{- 0.1 \cdot 30}{30} - \frac{11}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.4

Combine the numerators over the common denominator.

$- 0.2 y = \frac{- 0.1 \cdot 30 - 11}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $- 0.1$ by $30$ .

$- 0.2 y = \frac{- 3 - 11}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.5.2

Subtract $11$ from $- 3$ .

$- 0.2 y = \frac{- 14}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$- 0.2 y = \frac{- 14}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.6

Cancel the common factor of $- 14$ and $30$ .

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Step 3.1.6.1

Factor $2$ out of $- 14$ .

$- 0.2 y = \frac{2 (- 7)}{30}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.6.2

Cancel the common factors.

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Step 3.1.6.2.1

Factor $2$ out of $30$ .

$- 0.2 y = \frac{2 \cdot - 7}{2 \cdot 15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.6.2.2

Cancel the common factor.

$- 0.2 y = \frac{2 \cdot - 7}{2 \cdot 15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.6.2.3

Rewrite the expression.

$- 0.2 y = \frac{- 7}{15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$- 0.2 y = \frac{- 7}{15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$- 0.2 y = \frac{- 7}{15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.1.7

Move the negative in front of the fraction.

$- 0.2 y = - \frac{7}{15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$- 0.2 y = - \frac{7}{15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2

Divide each term in $- 0.2 y = - \frac{7}{15}$ by $- 0.2$ and simplify.

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Step 3.2.1

Divide each term in $- 0.2 y = - \frac{7}{15}$ by $- 0.2$ .

$\frac{- 0.2 y}{- 0.2} = \frac{- \frac{7}{15}}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 0.2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 0.2 y}{- 0.2} = \frac{- \frac{7}{15}}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{7}{15}}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{- \frac{7}{15}}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{- \frac{7}{15}}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{7}{15} \cdot \frac{1}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.2

Cancel the common factor of $0.2$ .

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Step 3.2.3.2.1

Move the leading negative in $- \frac{7}{15}$ into the numerator.

$y = \frac{- 7}{15} \cdot \frac{1}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.2.2

Factor $0.2$ out of $- 7$ .

$y = \frac{0.2 (- 35)}{15} \cdot \frac{1}{- 0.2}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.2.3

Factor $0.2$ out of $- 0.2$ .

$y = \frac{0.2 \cdot - 35}{15} \cdot \frac{1}{0.2 \cdot - 1}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.2.4

Cancel the common factor.

$y = \frac{0.2 \cdot - 35}{15} \cdot \frac{1}{0.2 \cdot - 1}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.2.5

Rewrite the expression.

$y = \frac{- 35}{15} \cdot \frac{1}{- 1}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{- 35}{15} \cdot \frac{1}{- 1}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.3

Multiply $\frac{- 35}{15}$ by $\frac{1}{- 1}$ .

$y = \frac{- 35}{15 \cdot - 1}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.4

Multiply $15$ by $- 1$ .

$y = \frac{- 35}{- 15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.5

Cancel the common factor of $- 35$ and $- 15$ .

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Step 3.2.3.5.1

Factor $- 5$ out of $- 35$ .

$y = \frac{- 5 \cdot 7}{- 15}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.5.2

Cancel the common factors.

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Step 3.2.3.5.2.1

Factor $- 5$ out of $- 15$ .

$y = \frac{- 5 \cdot 7}{- 5 \cdot 3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.5.2.2

Cancel the common factor.

$y = \frac{- 5 \cdot 7}{- 5 \cdot 3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 3.2.3.5.2.3

Rewrite the expression.

$y = \frac{7}{3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{7}{3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{7}{3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{7}{3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{7}{3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

$y = \frac{7}{3}$

$\frac{11}{6} - 0.4 y = 0.9$

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

Step 4

Replace all occurrences of $y$ with $\frac{7}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $\frac{11}{6} - 0.4 y = 0.9$ with $\frac{7}{3}$ .

$\frac{11}{6} - 0.4 (\frac{7}{3}) = 0.9$

$y = \frac{7}{3}$

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\frac{11}{6} - 0.4 (\frac{7}{3})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $- 0.4 (\frac{7}{3})$ .

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Step 4.2.1.1.1.1

Combine $- 0.4$ and $\frac{7}{3}$ .

$\frac{11}{6} + \frac{- 0.4 \cdot 7}{3} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.1.1.2

Multiply $- 0.4$ by $7$ .

$\frac{11}{6} + \frac{- 2.8}{3} = 0.9$

$y = \frac{7}{3}$

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

$\frac{11}{6} + \frac{- 2.8}{3} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.1.2

Divide $- 2.8$ by $3$ .

$\frac{11}{6} - 0.9\overline{3} = 0.9$

$y = \frac{7}{3}$

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

$\frac{11}{6} - 0.9\overline{3} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.2

To write $- 0.9\overline{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{11}{6} - 0.9\overline{3} \cdot \frac{6}{6} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.3

Combine fractions.

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Step 4.2.1.3.1

Combine $- 0.9\overline{3}$ and $\frac{6}{6}$ .

$\frac{11}{6} + \frac{- 0.9\overline{3} \cdot 6}{6} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.3.2

Combine the numerators over the common denominator.

$\frac{11 - 0.9\overline{3} \cdot 6}{6} = 0.9$

$y = \frac{7}{3}$

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

$\frac{11 - 0.9\overline{3} \cdot 6}{6} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.4

Simplify the numerator.

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Step 4.2.1.4.1

Multiply $- 0.9\overline{3}$ by $6$ .

$\frac{11 - 5.5\overline{9}}{6} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.4.2

Subtract $5.5\overline{9}$ from $11$ .

$\frac{5.4}{6} = 0.9$

$y = \frac{7}{3}$

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

$\frac{5.4}{6} = 0.9$

$y = \frac{7}{3}$

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

Step 4.2.1.5

Divide $5.4$ by $6$ .

$0.9 = 0.9$

$y = \frac{7}{3}$

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

$0.9 = 0.9$

$y = \frac{7}{3}$

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

$0.9 = 0.9$

$y = \frac{7}{3}$

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

Step 4.3

Replace all occurrences of $y$ in $x = \frac{11}{3} - y$ with $\frac{7}{3}$ .

$x = \frac{11}{3} - (\frac{7}{3})$

$0.9 = 0.9$

$y = \frac{7}{3}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $\frac{11}{3} - (\frac{7}{3})$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{11 - 7}{3}$

$0.9 = 0.9$

$y = \frac{7}{3}$

Step 4.4.1.2

Subtract $7$ from $11$ .

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

$0.9 = 0.9$

$y = \frac{7}{3}$

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

$0.9 = 0.9$

$y = \frac{7}{3}$

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

$0.9 = 0.9$

$y = \frac{7}{3}$

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

$0.9 = 0.9$

$y = \frac{7}{3}$

Step 5

Remove any equations from the system that are always true.

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

$y = \frac{7}{3}$

Step 6

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

$(\frac{4}{3}, \frac{7}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{3}, \frac{7}{3})$

Equation Form:

$x = \frac{4}{3}, y = \frac{7}{3}$

Step 8