Algebra Examples

$- 7 m = 2 (3 n + 13)$ $7 n = 2 (m - 5)$

Step 1

Divide each term in $- 7 m = 2 (3 n + 13)$ by $- 7$ and simplify.

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Step 1.1

Divide each term in $- 7 m = 2 (3 n + 13)$ by $- 7$ .

$\frac{- 7 m}{- 7} = \frac{2 (3 n + 13)}{- 7}$

$7 n = 2 (m - 5)$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 7$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 7 m}{- 7} = \frac{2 (3 n + 13)}{- 7}$

$7 n = 2 (m - 5)$

Step 1.2.1.2

Divide $m$ by $1$ .

$m = \frac{2 (3 n + 13)}{- 7}$

$7 n = 2 (m - 5)$

$m = \frac{2 (3 n + 13)}{- 7}$

$7 n = 2 (m - 5)$

$m = \frac{2 (3 n + 13)}{- 7}$

$7 n = 2 (m - 5)$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = 2 (m - 5)$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = 2 (m - 5)$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = 2 (m - 5)$

Step 2

Replace all occurrences of $m$ with $- \frac{2 (3 n + 13)}{7}$ in each equation.

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Step 2.1

Replace all occurrences of $m$ in $7 n = 2 (m - 5)$ with $- \frac{2 (3 n + 13)}{7}$ .

$7 n = 2 ((- \frac{2 (3 n + 13)}{7}) - 5)$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 ((- \frac{2 (3 n + 13)}{7}) - 5)$ .

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Step 2.2.1.1

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

$7 n = 2 (- \frac{2 (3 n + 13)}{7} - 5 \cdot \frac{7}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.2

Simplify terms.

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Step 2.2.1.2.1

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

$7 n = 2 (- \frac{2 (3 n + 13)}{7} + \frac{- 5 \cdot 7}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.2.2

Combine the numerators over the common denominator.

$7 n = 2 (\frac{- 2 (3 n + 13) - 5 \cdot 7}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = 2 (\frac{- 2 (3 n + 13) - 5 \cdot 7}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3

Simplify the numerator.

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Step 2.2.1.3.1

Factor $- 1$ out of $- 2 (3 n + 13) - 5 \cdot 7$ .

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Step 2.2.1.3.1.1

Factor $- 1$ out of $- 2 (3 n + 13)$ .

$7 n = 2 (\frac{- (2 (3 n + 13)) - 5 \cdot 7}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.1.2

Factor $- 1$ out of $- 5 \cdot 7$ .

$7 n = 2 (\frac{- (2 (3 n + 13)) - (5 \cdot 7)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.1.3

Factor $- 1$ out of $- (2 (3 n + 13)) - (5 \cdot 7)$ .

$7 n = 2 (\frac{- (2 (3 n + 13) + 5 \cdot 7)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = 2 (\frac{- (2 (3 n + 13) + 5 \cdot 7)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.2

Apply the distributive property.

$7 n = 2 (\frac{- (2 (3 n) + 2 \cdot 13 + 5 \cdot 7)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.3

Multiply $3$ by $2$ .

$7 n = 2 (\frac{- (6 n + 2 \cdot 13 + 5 \cdot 7)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.4

Multiply $2$ by $13$ .

$7 n = 2 (\frac{- (6 n + 26 + 5 \cdot 7)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.5

Multiply $5$ by $7$ .

$7 n = 2 (\frac{- (6 n + 26 + 35)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.3.6

Add $26$ and $35$ .

$7 n = 2 (\frac{- (6 n + 61)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = 2 (\frac{- (6 n + 61)}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.4

Move the negative in front of the fraction.

$7 n = 2 (- \frac{6 n + 61}{7})$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.5

Multiply $2 (- \frac{6 n + 61}{7})$ .

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Step 2.2.1.5.1

Multiply $- 1$ by $2$ .

$7 n = - 2 \frac{6 n + 61}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.5.2

Combine $- 2$ and $\frac{6 n + 61}{7}$ .

$7 n = \frac{- 2 (6 n + 61)}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = \frac{- 2 (6 n + 61)}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

Step 2.2.1.6

Move the negative in front of the fraction.

$7 n = - \frac{2 (6 n + 61)}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = - \frac{2 (6 n + 61)}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = - \frac{2 (6 n + 61)}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

$7 n = - \frac{2 (6 n + 61)}{7}$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3

Solve for $n$ in $7 n = - \frac{2 (6 n + 61)}{7}$ .

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Step 3.1

Multiply both sides by $7$ .

$7 n \cdot 7 = - \frac{2 (6 n + 61)}{7} \cdot 7$

$m = - \frac{2 (3 n + 13)}{7}$

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

Multiply $7$ by $7$ .

$49 n = - \frac{2 (6 n + 61)}{7} \cdot 7$

$m = - \frac{2 (3 n + 13)}{7}$

$49 n = - \frac{2 (6 n + 61)}{7} \cdot 7$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{2 (6 n + 61)}{7} \cdot 7$ .

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Step 3.2.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{2 (6 n + 61)}{7}$ into the numerator.

$49 n = \frac{- 2 (6 n + 61)}{7} \cdot 7$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.2.2.1.1.2

Cancel the common factor.

$49 n = \frac{- 2 (6 n + 61)}{7} \cdot 7$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.2.2.1.1.3

Rewrite the expression.

$49 n = - 2 (6 n + 61)$

$m = - \frac{2 (3 n + 13)}{7}$

$49 n = - 2 (6 n + 61)$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.2.2.1.2

Apply the distributive property.

$49 n = - 2 (6 n) - 2 \cdot 61$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.2.2.1.3

Multiply.

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Step 3.2.2.1.3.1

Multiply $6$ by $- 2$ .

$49 n = - 12 n - 2 \cdot 61$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.2.2.1.3.2

Multiply $- 2$ by $61$ .

$49 n = - 12 n - 122$

$m = - \frac{2 (3 n + 13)}{7}$

$49 n = - 12 n - 122$

$m = - \frac{2 (3 n + 13)}{7}$

$49 n = - 12 n - 122$

$m = - \frac{2 (3 n + 13)}{7}$

$49 n = - 12 n - 122$

$m = - \frac{2 (3 n + 13)}{7}$

$49 n = - 12 n - 122$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.3

Solve for $n$ .

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Step 3.3.1

Move all terms containing $n$ to the left side of the equation.

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Step 3.3.1.1

Add $12 n$ to both sides of the equation.

$49 n + 12 n = - 122$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.3.1.2

Add $49 n$ and $12 n$ .

$61 n = - 122$

$m = - \frac{2 (3 n + 13)}{7}$

$61 n = - 122$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.3.2

Divide each term in $61 n = - 122$ by $61$ and simplify.

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Step 3.3.2.1

Divide each term in $61 n = - 122$ by $61$ .

$\frac{61 n}{61} = \frac{- 122}{61}$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $61$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{61 n}{61} = \frac{- 122}{61}$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.3.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 122}{61}$

$m = - \frac{2 (3 n + 13)}{7}$

$n = \frac{- 122}{61}$

$m = - \frac{2 (3 n + 13)}{7}$

$n = \frac{- 122}{61}$

$m = - \frac{2 (3 n + 13)}{7}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 122$ by $61$ .

$n = - 2$

$m = - \frac{2 (3 n + 13)}{7}$

$n = - 2$

$m = - \frac{2 (3 n + 13)}{7}$

$n = - 2$

$m = - \frac{2 (3 n + 13)}{7}$

$n = - 2$

$m = - \frac{2 (3 n + 13)}{7}$

$n = - 2$

$m = - \frac{2 (3 n + 13)}{7}$

Step 4

Replace all occurrences of $n$ with $- 2$ in each equation.

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Step 4.1

Replace all occurrences of $n$ in $m = - \frac{2 (3 n + 13)}{7}$ with $- 2$ .

$m = - \frac{2 (3 (- 2) + 13)}{7}$

$n = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{2 (3 (- 2) + 13)}{7}$ .

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Step 4.2.1.1

Simplify the numerator.

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Step 4.2.1.1.1

Multiply $3$ by $- 2$ .

$m = - \frac{2 (- 6 + 13)}{7}$

$n = - 2$

Step 4.2.1.1.2

Add $- 6$ and $13$ .

$m = - \frac{2 \cdot 7}{7}$

$n = - 2$

$m = - \frac{2 \cdot 7}{7}$

$n = - 2$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $2$ by $7$ .

$m = - \frac{14}{7}$

$n = - 2$

Step 4.2.1.2.2

Divide $14$ by $7$ .

$m = - 1 \cdot 2$

$n = - 2$

Step 4.2.1.2.3

Multiply $- 1$ by $2$ .

$m = - 2$

$n = - 2$

$m = - 2$

$n = - 2$

$m = - 2$

$n = - 2$

$m = - 2$

$n = - 2$

$m = - 2$

$n = - 2$

Step 5

List all of the solutions.

$m = - 2, n = - 2$