Algebra Examples

$\frac{1}{m} + \frac{1}{n} = 5$ $\frac{2}{m} + \frac{3}{n} = 12$

Step 1

Solve for $m$ in $\frac{1}{m} + \frac{1}{n} = 5$ .

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Step 1.1

Subtract $\frac{1}{n}$ from both sides of the equation.

$\frac{1}{m} = 5 - \frac{1}{n}$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2

Find the LCD of the terms in the equation.

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Step 1.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$m, 1, n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.2

Since $m, 1, n$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $m^{1}, n^{1}$ .

Since $m, 1, n$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part m,n.

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

$1$ List the prime factors of each number.

$2$ Multiply each factor the greatest number of times it occurs in either number.

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.6

The factor for $m^{1}$ is $m$ itself.

$m = m$

m occurs $1$ time.

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.7

The factor for $n^{1}$ is $n$ itself.

$n = n$

n occurs $1$ time.

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.8

The LCM of $m^{1}, n^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$m \cdot n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.2.9

Multiply $m$ by $n$ .

$m n$

$\frac{2}{m} + \frac{3}{n} = 12$

$m n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3

Multiply each term in $\frac{1}{m} = 5 - \frac{1}{n}$ by $m n$ to eliminate the fractions.

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Step 1.3.1

Multiply each term in $\frac{1}{m} = 5 - \frac{1}{n}$ by $m n$ .

$\frac{1}{m} \cdot (m n) = 5 (m n) - \frac{1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $m$ .

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Step 1.3.2.1.1

Factor $m$ out of $m n$ .

$\frac{1}{m} \cdot (m (n)) = 5 (m n) - \frac{1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.2.1.2

Cancel the common factor.

$\frac{1}{m} \cdot (m n) = 5 (m n) - \frac{1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.2.1.3

Rewrite the expression.

$n = 5 (m n) - \frac{1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

$n = 5 (m n) - \frac{1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

$n = 5 (m n) - \frac{1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Cancel the common factor of $n$ .

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Step 1.3.3.1.1

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

$n = 5 m n + \frac{- 1}{n} \cdot (m n)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.3.1.2

Factor $n$ out of $m n$ .

$n = 5 m n + \frac{- 1}{n} \cdot (n m)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.3.1.3

Cancel the common factor.

$n = 5 m n + \frac{- 1}{n} \cdot (n m)$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.3.3.1.4

Rewrite the expression.

$n = 5 m n - m$

$\frac{2}{m} + \frac{3}{n} = 12$

$n = 5 m n - m$

$\frac{2}{m} + \frac{3}{n} = 12$

$n = 5 m n - m$

$\frac{2}{m} + \frac{3}{n} = 12$

$n = 5 m n - m$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4

Solve the equation.

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Step 1.4.1

Rewrite the equation as $5 m n - m = n$ .

$5 m n - m = n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4.2

Factor $m$ out of $5 m n - m$ .

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Step 1.4.2.1

Factor $m$ out of $5 m n$ .

$m (5 n) - m = n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4.2.2

Factor $m$ out of $- m$ .

$m (5 n) + m \cdot - 1 = n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4.2.3

Factor $m$ out of $m (5 n) + m \cdot - 1$ .

$m (5 n - 1) = n$

$\frac{2}{m} + \frac{3}{n} = 12$

$m (5 n - 1) = n$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4.3

Divide each term in $m (5 n - 1) = n$ by $5 n - 1$ and simplify.

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Step 1.4.3.1

Divide each term in $m (5 n - 1) = n$ by $5 n - 1$ .

$\frac{m (5 n - 1)}{5 n - 1} = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4.3.2

Simplify the left side.

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Step 1.4.3.2.1

Cancel the common factor of $5 n - 1$ .

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Step 1.4.3.2.1.1

Cancel the common factor.

$\frac{m (5 n - 1)}{5 n - 1} = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 1.4.3.2.1.2

Divide $m$ by $1$ .

$m = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{2}{m} + \frac{3}{n} = 12$

Step 2

Replace all occurrences of $m$ with $\frac{n}{5 n - 1}$ in each equation.

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Step 2.1

Replace all occurrences of $m$ in $\frac{2}{m} + \frac{3}{n} = 12$ with $\frac{n}{5 n - 1}$ .

$\frac{2}{\frac{n}{5 n - 1}} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$2 (\frac{5 n - 1}{n}) + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2.1.1.2

Combine $2$ and $\frac{5 n - 1}{n}$ .

$\frac{2 (5 n - 1)}{n} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{2 (5 n - 1)}{n} + \frac{3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2.1.2

Combine the numerators over the common denominator.

$\frac{2 (5 n - 1) + 3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2.1.3

Simplify each term.

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Step 2.2.1.3.1

Apply the distributive property.

$\frac{2 (5 n) + 2 \cdot - 1 + 3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2.1.3.2

Multiply $5$ by $2$ .

$\frac{10 n + 2 \cdot - 1 + 3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2.1.3.3

Multiply $2$ by $- 1$ .

$\frac{10 n - 2 + 3}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{10 n - 2 + 3}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 2.2.1.4

Add $- 2$ and $3$ .

$\frac{10 n + 1}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{10 n + 1}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{10 n + 1}{n} = 12$

$m = \frac{n}{5 n - 1}$

$\frac{10 n + 1}{n} = 12$

$m = \frac{n}{5 n - 1}$

Step 3

Solve for $n$ in $\frac{10 n + 1}{n} = 12$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$n, 1$

$m = \frac{n}{5 n - 1}$

Step 3.1.2

The LCM of one and any expression is the expression.

$m = \frac{n}{5 n - 1}$

Step 3.2

Multiply each term in $\frac{10 n + 1}{n} = 12$ by $n$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{10 n + 1}{n} = 12$ by $n$ .

$\frac{10 n + 1}{n} \cdot n = 12 n$

$m = \frac{n}{5 n - 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $n$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 n + 1}{n} \cdot n = 12 n$

$m = \frac{n}{5 n - 1}$

Step 3.2.2.1.2

Rewrite the expression.

$10 n + 1 = 12 n$

$m = \frac{n}{5 n - 1}$

$10 n + 1 = 12 n$

$m = \frac{n}{5 n - 1}$

$10 n + 1 = 12 n$

$m = \frac{n}{5 n - 1}$

$10 n + 1 = 12 n$

$m = \frac{n}{5 n - 1}$

Step 3.3

Solve the equation.

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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

Subtract $12 n$ from both sides of the equation.

$10 n + 1 - 12 n = 0$

$m = \frac{n}{5 n - 1}$

Step 3.3.1.2

Subtract $12 n$ from $10 n$ .

$- 2 n + 1 = 0$

$m = \frac{n}{5 n - 1}$

$- 2 n + 1 = 0$

$m = \frac{n}{5 n - 1}$

Step 3.3.2

Subtract $1$ from both sides of the equation.

$- 2 n = - 1$

$m = \frac{n}{5 n - 1}$

Step 3.3.3

Divide each term in $- 2 n = - 1$ by $- 2$ and simplify.

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Step 3.3.3.1

Divide each term in $- 2 n = - 1$ by $- 2$ .

$\frac{- 2 n}{- 2} = \frac{- 1}{- 2}$

$m = \frac{n}{5 n - 1}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 n}{- 2} = \frac{- 1}{- 2}$

$m = \frac{n}{5 n - 1}$

Step 3.3.3.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 1}{- 2}$

$m = \frac{n}{5 n - 1}$

$n = \frac{- 1}{- 2}$

$m = \frac{n}{5 n - 1}$

$n = \frac{- 1}{- 2}$

$m = \frac{n}{5 n - 1}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$n = \frac{1}{2}$

$m = \frac{n}{5 n - 1}$

$n = \frac{1}{2}$

$m = \frac{n}{5 n - 1}$

$n = \frac{1}{2}$

$m = \frac{n}{5 n - 1}$

$n = \frac{1}{2}$

$m = \frac{n}{5 n - 1}$

$n = \frac{1}{2}$

$m = \frac{n}{5 n - 1}$

Step 4

Replace all occurrences of $n$ with $\frac{1}{2}$ in each equation.

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Step 4.1

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

$m = \frac{\frac{1}{2}}{5 (\frac{1}{2}) - 1}$

$n = \frac{1}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$m = \frac{1}{2} \cdot \frac{1}{5 (\frac{1}{2}) - 1}$

$n = \frac{1}{2}$

Step 4.2.1.2

Simplify the denominator.

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Step 4.2.1.2.1

Combine $5$ and $\frac{1}{2}$ .

$m = \frac{1}{2} \cdot \frac{1}{\frac{5}{2} - 1}$

$n = \frac{1}{2}$

Step 4.2.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$m = \frac{1}{2} \cdot \frac{1}{\frac{5}{2} - 1 \cdot \frac{2}{2}}$

$n = \frac{1}{2}$

Step 4.2.1.2.3

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

$m = \frac{1}{2} \cdot \frac{1}{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}$

$n = \frac{1}{2}$

Step 4.2.1.2.4

Combine the numerators over the common denominator.

$m = \frac{1}{2} \cdot \frac{1}{\frac{5 - 1 \cdot 2}{2}}$

$n = \frac{1}{2}$

Step 4.2.1.2.5

Simplify the numerator.

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Step 4.2.1.2.5.1

Multiply $- 1$ by $2$ .

$m = \frac{1}{2} \cdot \frac{1}{\frac{5 - 2}{2}}$

$n = \frac{1}{2}$

Step 4.2.1.2.5.2

Subtract $2$ from $5$ .

$m = \frac{1}{2} \cdot \frac{1}{\frac{3}{2}}$

$n = \frac{1}{2}$

$m = \frac{1}{2} \cdot \frac{1}{\frac{3}{2}}$

$n = \frac{1}{2}$

$m = \frac{1}{2} \cdot \frac{1}{\frac{3}{2}}$

$n = \frac{1}{2}$

Step 4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$m = \frac{1}{2} \cdot (1 (\frac{2}{3}))$

$n = \frac{1}{2}$

Step 4.2.1.4

Multiply $\frac{2}{3}$ by $1$ .

$m = \frac{1}{2} \cdot \frac{2}{3}$

$n = \frac{1}{2}$

Step 4.2.1.5

Cancel the common factor of $2$ .

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Step 4.2.1.5.1

Cancel the common factor.

$m = \frac{1}{2} \cdot \frac{2}{3}$

$n = \frac{1}{2}$

Step 4.2.1.5.2

Rewrite the expression.

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

$n = \frac{1}{2}$

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

$n = \frac{1}{2}$

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

$n = \frac{1}{2}$

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

$n = \frac{1}{2}$

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

$n = \frac{1}{2}$

Step 5

List all of the solutions.

$m = \frac{1}{3}, n = \frac{1}{2}$