Finite Math Examples

$3 m + 3 n = - 1$ , $2 m + 5 n = 3$

Step 1

Solve for $m$ in $3 m + 3 n = - 1$ .

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Step 1.1

Subtract $3 n$ from both sides of the equation.

$3 m = - 1 - 3 n$

$2 m + 5 n = 3$

Step 1.2

Divide each term in $3 m = - 1 - 3 n$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 m = - 1 - 3 n$ by $3$ .

$\frac{3 m}{3} = \frac{- 1}{3} + \frac{- 3 n}{3}$

$2 m + 5 n = 3$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 m}{3} = \frac{- 1}{3} + \frac{- 3 n}{3}$

$2 m + 5 n = 3$

Step 1.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 1}{3} + \frac{- 3 n}{3}$

$2 m + 5 n = 3$

$m = \frac{- 1}{3} + \frac{- 3 n}{3}$

$2 m + 5 n = 3$

$m = \frac{- 1}{3} + \frac{- 3 n}{3}$

$2 m + 5 n = 3$

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.

$m = - \frac{1}{3} + \frac{- 3 n}{3}$

$2 m + 5 n = 3$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

Factor $3$ out of $- 3 n$ .

$m = - \frac{1}{3} + \frac{3 (- n)}{3}$

$2 m + 5 n = 3$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $3$ .

$m = - \frac{1}{3} + \frac{3 (- n)}{3 (1)}$

$2 m + 5 n = 3$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$m = - \frac{1}{3} + \frac{3 (- n)}{3 \cdot 1}$

$2 m + 5 n = 3$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$m = - \frac{1}{3} + \frac{- n}{1}$

$2 m + 5 n = 3$

Step 1.2.3.1.2.2.4

Divide $- n$ by $1$ .

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

$m = - \frac{1}{3} - n$

$2 m + 5 n = 3$

Step 2

Replace all occurrences of $m$ with $- \frac{1}{3} - n$ in each equation.

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Step 2.1

Replace all occurrences of $m$ in $2 m + 5 n = 3$ with $- \frac{1}{3} - n$ .

$2 (- \frac{1}{3} - n) + 5 n = 3$

$m = - \frac{1}{3} - n$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- \frac{1}{3} - n) + 5 n$ .

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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.

$2 (- \frac{1}{3}) + 2 (- n) + 5 n = 3$

$m = - \frac{1}{3} - n$

Step 2.2.1.1.2

Multiply $2 (- \frac{1}{3})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $2$ .

$- 2 (\frac{1}{3}) + 2 (- n) + 5 n = 3$

$m = - \frac{1}{3} - n$

Step 2.2.1.1.2.2

Combine $- 2$ and $\frac{1}{3}$ .

$\frac{- 2}{3} + 2 (- n) + 5 n = 3$

$m = - \frac{1}{3} - n$

$\frac{- 2}{3} + 2 (- n) + 5 n = 3$

$m = - \frac{1}{3} - n$

Step 2.2.1.1.3

Multiply $- 1$ by $2$ .

$\frac{- 2}{3} - 2 n + 5 n = 3$

$m = - \frac{1}{3} - n$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{2}{3} - 2 n + 5 n = 3$

$m = - \frac{1}{3} - n$

$- \frac{2}{3} - 2 n + 5 n = 3$

$m = - \frac{1}{3} - n$

Step 2.2.1.2

Add $- 2 n$ and $5 n$ .

$- \frac{2}{3} + 3 n = 3$

$m = - \frac{1}{3} - n$

$- \frac{2}{3} + 3 n = 3$

$m = - \frac{1}{3} - n$

$- \frac{2}{3} + 3 n = 3$

$m = - \frac{1}{3} - n$

$- \frac{2}{3} + 3 n = 3$

$m = - \frac{1}{3} - n$

Step 3

Solve for $n$ in $- \frac{2}{3} + 3 n = 3$ .

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Step 3.1

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

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Step 3.1.1

Add $\frac{2}{3}$ to both sides of the equation.

$3 n = 3 + \frac{2}{3}$

$m = - \frac{1}{3} - n$

Step 3.1.2

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

$3 n = 3 \cdot \frac{3}{3} + \frac{2}{3}$

$m = - \frac{1}{3} - n$

Step 3.1.3

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

$3 n = \frac{3 \cdot 3}{3} + \frac{2}{3}$

$m = - \frac{1}{3} - n$

Step 3.1.4

Combine the numerators over the common denominator.

$3 n = \frac{3 \cdot 3 + 2}{3}$

$m = - \frac{1}{3} - n$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $3$ by $3$ .

$3 n = \frac{9 + 2}{3}$

$m = - \frac{1}{3} - n$

Step 3.1.5.2

Add $9$ and $2$ .

$3 n = \frac{11}{3}$

$m = - \frac{1}{3} - n$

$3 n = \frac{11}{3}$

$m = - \frac{1}{3} - n$

$3 n = \frac{11}{3}$

$m = - \frac{1}{3} - n$

Step 3.2

Divide each term in $3 n = \frac{11}{3}$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 n = \frac{11}{3}$ by $3$ .

$\frac{3 n}{3} = \frac{\frac{11}{3}}{3}$

$m = - \frac{1}{3} - n$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 n}{3} = \frac{\frac{11}{3}}{3}$

$m = - \frac{1}{3} - n$

Step 3.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{\frac{11}{3}}{3}$

$m = - \frac{1}{3} - n$

$n = \frac{\frac{11}{3}}{3}$

$m = - \frac{1}{3} - n$

$n = \frac{\frac{11}{3}}{3}$

$m = - \frac{1}{3} - n$

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.

$n = \frac{11}{3} \cdot \frac{1}{3}$

$m = - \frac{1}{3} - n$

Step 3.2.3.2

Multiply $\frac{11}{3} \cdot \frac{1}{3}$ .

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Step 3.2.3.2.1

Multiply $\frac{11}{3}$ by $\frac{1}{3}$ .

$n = \frac{11}{3 \cdot 3}$

$m = - \frac{1}{3} - n$

Step 3.2.3.2.2

Multiply $3$ by $3$ .

$n = \frac{11}{9}$

$m = - \frac{1}{3} - n$

$n = \frac{11}{9}$

$m = - \frac{1}{3} - n$

$n = \frac{11}{9}$

$m = - \frac{1}{3} - n$

$n = \frac{11}{9}$

$m = - \frac{1}{3} - n$

$n = \frac{11}{9}$

$m = - \frac{1}{3} - n$

Step 4

Replace all occurrences of $n$ with $\frac{11}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $n$ in $m = - \frac{1}{3} - n$ with $\frac{11}{9}$ .

$m = - \frac{1}{3} - (\frac{11}{9})$

$n = \frac{11}{9}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{1}{3} - (\frac{11}{9})$ .

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Step 4.2.1.1

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

$m = - \frac{1}{3} \cdot \frac{3}{3} - \frac{11}{9}$

$n = \frac{11}{9}$

Step 4.2.1.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.2.1

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$m = - \frac{3}{3 \cdot 3} - \frac{11}{9}$

$n = \frac{11}{9}$

Step 4.2.1.2.2

Multiply $3$ by $3$ .

$m = - \frac{3}{9} - \frac{11}{9}$

$n = \frac{11}{9}$

$m = - \frac{3}{9} - \frac{11}{9}$

$n = \frac{11}{9}$

Step 4.2.1.3

Combine the numerators over the common denominator.

$m = \frac{- 3 - 11}{9}$

$n = \frac{11}{9}$

Step 4.2.1.4

Subtract $11$ from $- 3$ .

$m = \frac{- 14}{9}$

$n = \frac{11}{9}$

Step 4.2.1.5

Move the negative in front of the fraction.

$m = - \frac{14}{9}$

$n = \frac{11}{9}$

$m = - \frac{14}{9}$

$n = \frac{11}{9}$

$m = - \frac{14}{9}$

$n = \frac{11}{9}$

$m = - \frac{14}{9}$

$n = \frac{11}{9}$

Step 5

List all of the solutions.

$m = - \frac{14}{9}, n = \frac{11}{9}$