Finite Math Examples

$7 x + 2 y = - 9$ , $- 2 x + 3 y + 5 = 6$

Step 1

Solve for $x$ in $7 x + 2 y = - 9$ .

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Step 1.1

Subtract $2 y$ from both sides of the equation.

$7 x = - 9 - 2 y$

$- 2 x + 3 y + 5 = 6$

Step 1.2

Divide each term in $7 x = - 9 - 2 y$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 x = - 9 - 2 y$ by $7$ .

$\frac{7 x}{7} = \frac{- 9}{7} + \frac{- 2 y}{7}$

$- 2 x + 3 y + 5 = 6$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 9}{7} + \frac{- 2 y}{7}$

$- 2 x + 3 y + 5 = 6$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{7} + \frac{- 2 y}{7}$

$- 2 x + 3 y + 5 = 6$

$x = \frac{- 9}{7} + \frac{- 2 y}{7}$

$- 2 x + 3 y + 5 = 6$

$x = \frac{- 9}{7} + \frac{- 2 y}{7}$

$- 2 x + 3 y + 5 = 6$

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.

$x = - \frac{9}{7} + \frac{- 2 y}{7}$

$- 2 x + 3 y + 5 = 6$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{9}{7} - \frac{2 y}{7}$

$- 2 x + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$- 2 x + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$- 2 x + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$- 2 x + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$- 2 x + 3 y + 5 = 6$

Step 2

Replace all occurrences of $x$ with $- \frac{9}{7} - \frac{2 y}{7}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 2 x + 3 y + 5 = 6$ with $- \frac{9}{7} - \frac{2 y}{7}$ .

$- 2 (- \frac{9}{7} - \frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 (- \frac{9}{7} - \frac{2 y}{7}) + 3 y + 5$ .

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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.

$- 2 (- \frac{9}{7}) - 2 (- \frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.1.2

Multiply $- 2 (- \frac{9}{7})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 2$ .

$2 (\frac{9}{7}) - 2 (- \frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.1.2.2

Combine $2$ and $\frac{9}{7}$ .

$\frac{2 \cdot 9}{7} - 2 (- \frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.1.2.3

Multiply $2$ by $9$ .

$\frac{18}{7} - 2 (- \frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{18}{7} - 2 (- \frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.1.3

Multiply $- 2 (- \frac{2 y}{7})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 2$ .

$\frac{18}{7} + 2 (\frac{2 y}{7}) + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.1.3.2

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

$\frac{18}{7} + \frac{2 (2 y)}{7} + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.1.3.3

Multiply $2$ by $2$ .

$\frac{18}{7} + \frac{4 y}{7} + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{18}{7} + \frac{4 y}{7} + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{18}{7} + \frac{4 y}{7} + 3 y + 5 = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{4 y}{7} + 3 y + \frac{18}{7} + 5 \cdot \frac{7}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.3

Combine $5$ and $\frac{7}{7}$ .

$\frac{4 y}{7} + 3 y + \frac{18}{7} + \frac{5 \cdot 7}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{4 y}{7} + 3 y + \frac{18 + 5 \cdot 7}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Multiply $5$ by $7$ .

$\frac{4 y}{7} + 3 y + \frac{18 + 35}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.5.2

Add $18$ and $35$ .

$\frac{4 y}{7} + 3 y + \frac{53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{4 y}{7} + 3 y + \frac{53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.6

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

$\frac{4 y}{7} + 3 y \cdot \frac{7}{7} + \frac{53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.7

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

$\frac{4 y}{7} + \frac{3 y \cdot 7}{7} + \frac{53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.8

Combine the numerators over the common denominator.

$\frac{4 y + 3 y \cdot 7}{7} + \frac{53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.9

Combine the numerators over the common denominator.

$\frac{4 y + 3 y \cdot 7 + 53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.10

Multiply $7$ by $3$ .

$\frac{4 y + 21 y + 53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 2.2.1.11

Add $4 y$ and $21 y$ .

$\frac{25 y + 53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{25 y + 53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{25 y + 53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$\frac{25 y + 53}{7} = 6$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3

Solve for $y$ in $\frac{25 y + 53}{7} = 6$ .

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Step 3.1

Multiply both sides by $7$ .

$\frac{25 y + 53}{7} \cdot 7 = 6 \cdot 7$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{25 y + 53}{7} \cdot 7 = 6 \cdot 7$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.2.1.1.2

Rewrite the expression.

$25 y + 53 = 6 \cdot 7$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$25 y + 53 = 6 \cdot 7$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$25 y + 53 = 6 \cdot 7$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $6$ by $7$ .

$25 y + 53 = 42$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$25 y + 53 = 42$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$25 y + 53 = 42$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $53$ from both sides of the equation.

$25 y = 42 - 53$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.3.1.2

Subtract $53$ from $42$ .

$25 y = - 11$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$25 y = - 11$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.3.2

Divide each term in $25 y = - 11$ by $25$ and simplify.

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Step 3.3.2.1

Divide each term in $25 y = - 11$ by $25$ .

$\frac{25 y}{25} = \frac{- 11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $25$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{25 y}{25} = \frac{- 11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$y = \frac{- 11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$y = \frac{- 11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$y = - \frac{11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$y = - \frac{11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$y = - \frac{11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

$y = - \frac{11}{25}$

$x = - \frac{9}{7} - \frac{2 y}{7}$

Step 4

Replace all occurrences of $y$ with $- \frac{11}{25}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{9}{7} - \frac{2 y}{7}$ with $- \frac{11}{25}$ .

$x = - \frac{9}{7} - \frac{2 (- \frac{11}{25})}{7}$

$y = - \frac{11}{25}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{9}{7} - \frac{2 (- \frac{11}{25})}{7}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 9 - 2 (- \frac{11}{25})}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.2

Multiply $- 2 (- \frac{11}{25})$ .

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Step 4.2.1.2.1

Multiply $- 1$ by $- 2$ .

$x = \frac{- 9 + 2 (\frac{11}{25})}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.2.2

Combine $2$ and $\frac{11}{25}$ .

$x = \frac{- 9 + \frac{2 \cdot 11}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.2.3

Multiply $2$ by $11$ .

$x = \frac{- 9 + \frac{22}{25}}{7}$

$y = - \frac{11}{25}$

$x = \frac{- 9 + \frac{22}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.3

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$x = \frac{- 9 \cdot \frac{25}{25} + \frac{22}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.4

Combine $- 9$ and $\frac{25}{25}$ .

$x = \frac{\frac{- 9 \cdot 25}{25} + \frac{22}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{- 9 \cdot 25 + 22}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $- 9$ by $25$ .

$x = \frac{\frac{- 225 + 22}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.6.2

Add $- 225$ and $22$ .

$x = \frac{\frac{- 203}{25}}{7}$

$y = - \frac{11}{25}$

$x = \frac{\frac{- 203}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.7

Move the negative in front of the fraction.

$x = \frac{- \frac{203}{25}}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{203}{25} \cdot \frac{1}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.9

Cancel the common factor of $7$ .

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Step 4.2.1.9.1

Move the leading negative in $- \frac{203}{25}$ into the numerator.

$x = \frac{- 203}{25} \cdot \frac{1}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.9.2

Factor $7$ out of $- 203$ .

$x = \frac{7 (- 29)}{25} \cdot \frac{1}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.9.3

Cancel the common factor.

$x = \frac{7 \cdot - 29}{25} \cdot \frac{1}{7}$

$y = - \frac{11}{25}$

Step 4.2.1.9.4

Rewrite the expression.

$x = \frac{- 29}{25}$

$y = - \frac{11}{25}$

$x = \frac{- 29}{25}$

$y = - \frac{11}{25}$

Step 4.2.1.10

Move the negative in front of the fraction.

$x = - \frac{29}{25}$

$y = - \frac{11}{25}$

$x = - \frac{29}{25}$

$y = - \frac{11}{25}$

$x = - \frac{29}{25}$

$y = - \frac{11}{25}$

$x = - \frac{29}{25}$

$y = - \frac{11}{25}$

Step 5

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

$(- \frac{29}{25}, - \frac{11}{25})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{29}{25}, - \frac{11}{25})$

Equation Form:

$x = - \frac{29}{25}, y = - \frac{11}{25}$

Step 7