Finite Math Examples

$- 9 a + 7 y = 3$ , $6 a - 7 y = 1$

Step 1

Solve for $a$ in $- 9 a + 7 y = 3$ .

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Step 1.1

Subtract $7 y$ from both sides of the equation.

$- 9 a = 3 - 7 y$

$6 a - 7 y = 1$

Step 1.2

Divide each term in $- 9 a = 3 - 7 y$ by $- 9$ and simplify.

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Step 1.2.1

Divide each term in $- 9 a = 3 - 7 y$ by $- 9$ .

$\frac{- 9 a}{- 9} = \frac{3}{- 9} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 9 a}{- 9} = \frac{3}{- 9} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{3}{- 9} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

$a = \frac{3}{- 9} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

$a = \frac{3}{- 9} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

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

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

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Step 1.2.3.1.1.1

Factor $3$ out of $3$ .

$a = \frac{3 (1)}{- 9} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $3$ out of $- 9$ .

$a = \frac{3 \cdot 1}{3 \cdot - 3} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$a = \frac{3 \cdot 1}{3 \cdot - 3} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$a = \frac{1}{- 3} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

$a = \frac{1}{- 3} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

$a = \frac{1}{- 3} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$a = - \frac{1}{3} + \frac{- 7 y}{- 9}$

$6 a - 7 y = 1$

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

$a = - \frac{1}{3} + \frac{7 y}{9}$

$6 a - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$6 a - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$6 a - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$6 a - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$6 a - 7 y = 1$

Step 2

Replace all occurrences of $a$ with $- \frac{1}{3} + \frac{7 y}{9}$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $6 a - 7 y = 1$ with $- \frac{1}{3} + \frac{7 y}{9}$ .

$6 (- \frac{1}{3} + \frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $6 (- \frac{1}{3} + \frac{7 y}{9}) - 7 y$ .

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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.

$6 (- \frac{1}{3}) + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$6 (\frac{- 1}{3}) + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.2.2

Factor $3$ out of $6$ .

$3 (2) (\frac{- 1}{3}) + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.2.3

Cancel the common factor.

$3 \cdot (2 (\frac{- 1}{3})) + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.2.4

Rewrite the expression.

$2 \cdot - 1 + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$2 \cdot - 1 + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.3

Multiply $2$ by $- 1$ .

$- 2 + 6 (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.1.4.1

Factor $3$ out of $6$ .

$- 2 + 3 (2) (\frac{7 y}{9}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.4.2

Factor $3$ out of $9$ .

$- 2 + 3 \cdot (2 (\frac{7 y}{3 \cdot 3})) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.4.3

Cancel the common factor.

$- 2 + 3 \cdot (2 (\frac{7 y}{3 \cdot 3})) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.4.4

Rewrite the expression.

$- 2 + 2 (\frac{7 y}{3}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 + 2 (\frac{7 y}{3}) - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.5

Combine $2$ and $\frac{7 y}{3}$ .

$- 2 + \frac{2 (7 y)}{3} - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.1.6

Multiply $7$ by $2$ .

$- 2 + \frac{14 y}{3} - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 + \frac{14 y}{3} - 7 y = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.2

To write $- 7 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 2 + \frac{14 y}{3} - 7 y \cdot \frac{3}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$- 2 + \frac{14 y}{3} + \frac{- 7 y \cdot 3}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- 2 + \frac{14 y - 7 y \cdot 3}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 + \frac{14 y - 7 y \cdot 3}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $7 y$ out of $14 y - 7 y \cdot 3$ .

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Step 2.2.1.4.1.1.1

Factor $7 y$ out of $14 y$ .

$- 2 + \frac{7 y (2) - 7 y \cdot 3}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4.1.1.2

Factor $7 y$ out of $- 7 y \cdot 3$ .

$- 2 + \frac{7 y (2) + 7 y (- 1 \cdot 3)}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4.1.1.3

Factor $7 y$ out of $7 y (2) + 7 y (- 1 \cdot 3)$ .

$- 2 + \frac{7 y (2 - 1 \cdot 3)}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 + \frac{7 y (2 - 1 \cdot 3)}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4.1.2

Multiply $- 1$ by $3$ .

$- 2 + \frac{7 y (2 - 3)}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4.1.3

Subtract $3$ from $2$ .

$- 2 + \frac{7 y \cdot - 1}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4.1.4

Multiply $- 1$ by $7$ .

$- 2 + \frac{- 7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 + \frac{- 7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$- 2 - \frac{7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 - \frac{7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 - \frac{7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 - \frac{7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$- 2 - \frac{7 y}{3} = 1$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3

Solve for $y$ in $- 2 - \frac{7 y}{3} = 1$ .

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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

Add $2$ to both sides of the equation.

$- \frac{7 y}{3} = 1 + 2$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.1.2

Add $1$ and $2$ .

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.2

Multiply both sides of the equation by $- \frac{3}{7}$ .

$- \frac{3}{7} \cdot (- \frac{7 y}{3}) = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$\frac{- 3}{7} \cdot (- \frac{7 y}{3}) = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{7 y}{3}$ into the numerator.

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{7} \cdot \frac{- 7 y}{3} = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{7} \cdot \frac{- 7 y}{3} = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{7} \cdot (- 7 y) = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$\frac{- 1}{7} \cdot (- 7 y) = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.2

Cancel the common factor of $7$ .

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Step 3.3.1.1.2.1

Factor $7$ out of $- 7 y$ .

$\frac{- 1}{7} \cdot (7 (- y)) = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{7} \cdot (7 (- y)) = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.2.3

Rewrite the expression.

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{7} \cdot 3$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{3}{7} \cdot 3$ .

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Step 3.3.2.1.1

Multiply $- \frac{3}{7} \cdot 3$ .

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Step 3.3.2.1.1.1

Multiply $3$ by $- 1$ .

$y = - 3 (\frac{3}{7})$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.2.1.1.2

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

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

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.2.1.1.3

Multiply $- 3$ by $3$ .

$y = \frac{- 9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$y = \frac{- 9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 3.3.2.1.2

Move the negative in front of the fraction.

$y = - \frac{9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$y = - \frac{9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$y = - \frac{9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$y = - \frac{9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

$y = - \frac{9}{7}$

$a = - \frac{1}{3} + \frac{7 y}{9}$

Step 4

Replace all occurrences of $y$ with $- \frac{9}{7}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $a = - \frac{1}{3} + \frac{7 y}{9}$ with $- \frac{9}{7}$ .

$a = - \frac{1}{3} + \frac{7 (- \frac{9}{7})}{9}$

$y = - \frac{9}{7}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{1}{3} + \frac{7 (- \frac{9}{7})}{9}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Simplify the numerator.

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Step 4.2.1.1.1.1

Multiply $- 1$ by $7$ .

$a = - \frac{1}{3} + \frac{- 7 (\frac{9}{7})}{9}$

$y = - \frac{9}{7}$

Step 4.2.1.1.1.2

Combine $- 7$ and $\frac{9}{7}$ .

$a = - \frac{1}{3} + \frac{\frac{- 7 \cdot 9}{7}}{9}$

$y = - \frac{9}{7}$

$a = - \frac{1}{3} + \frac{\frac{- 7 \cdot 9}{7}}{9}$

$y = - \frac{9}{7}$

Step 4.2.1.1.2

Multiply $- 7$ by $9$ .

$a = - \frac{1}{3} + \frac{\frac{- 63}{7}}{9}$

$y = - \frac{9}{7}$

Step 4.2.1.1.3

Divide $- 63$ by $7$ .

$a = - \frac{1}{3} + \frac{- 9}{9}$

$y = - \frac{9}{7}$

Step 4.2.1.1.4

Divide $- 9$ by $9$ .

$a = - \frac{1}{3} - 1$

$y = - \frac{9}{7}$

$a = - \frac{1}{3} - 1$

$y = - \frac{9}{7}$

Step 4.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - \frac{1}{3} - 1 \cdot \frac{3}{3}$

$y = - \frac{9}{7}$

Step 4.2.1.3

Combine $- 1$ and $\frac{3}{3}$ .

$a = - \frac{1}{3} + \frac{- 1 \cdot 3}{3}$

$y = - \frac{9}{7}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$a = \frac{- 1 - 1 \cdot 3}{3}$

$y = - \frac{9}{7}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 1$ by $3$ .

$a = \frac{- 1 - 3}{3}$

$y = - \frac{9}{7}$

Step 4.2.1.5.2

Subtract $3$ from $- 1$ .

$a = \frac{- 4}{3}$

$y = - \frac{9}{7}$

$a = \frac{- 4}{3}$

$y = - \frac{9}{7}$

Step 4.2.1.6

Move the negative in front of the fraction.

$a = - \frac{4}{3}$

$y = - \frac{9}{7}$

$a = - \frac{4}{3}$

$y = - \frac{9}{7}$

$a = - \frac{4}{3}$

$y = - \frac{9}{7}$

$a = - \frac{4}{3}$

$y = - \frac{9}{7}$

Step 5

List all of the solutions.

$a = - \frac{4}{3}, y = - \frac{9}{7}$