Algebra Examples

$\frac{x + m}{m} - \frac{x + n}{n} = \frac{m^{2} + n^{2}}{m n} - 2$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$m, n, m n, 1$

Step 1.2

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

Step 1.3

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.

Step 1.4

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

Not prime

Step 1.5

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

$1$

Step 1.6

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

$m^{1} = m$

$m$ occurs $1$ time.

Step 1.7

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

$n^{1} = n$

$n$ occurs $1$ time.

Step 1.8

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

$m^{1} = m$

$m$ occurs $1$ time.

Step 1.9

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

$n^{1} = n$

$n$ occurs $1$ time.

Step 1.10

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

$m \cdot n$

Step 1.11

Multiply $m$ by $n$ .

$m n$

Step 2

Multiply each term in $\frac{x + m}{m} - \frac{x + n}{n} = \frac{m^{2} + n^{2}}{m n} - 2$ by $m n$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x + m}{m} - \frac{x + n}{n} = \frac{m^{2} + n^{2}}{m n} - 2$ by $m n$ .

$\frac{x + m}{m} (m n) - \frac{x + n}{n} (m n) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $m$ .

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Step 2.2.1.1.1

Factor $m$ out of $m n$ .

$\frac{x + m}{m} (m (n)) - \frac{x + n}{n} (m n) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{x + m}{m} (m n) - \frac{x + n}{n} (m n) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.1.3

Rewrite the expression.

$(x + m) n - \frac{x + n}{n} (m n) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.2

Apply the distributive property.

$x n + m n - \frac{x + n}{n} (m n) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.3

Cancel the common factor of $n$ .

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Step 2.2.1.3.1

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

$x n + m n + \frac{- (x + n)}{n} (m n) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.3.2

Factor $n$ out of $m n$ .

$x n + m n + \frac{- (x + n)}{n} (n m) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.3.3

Cancel the common factor.

$x n + m n + \frac{- (x + n)}{n} (n m) = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.3.4

Rewrite the expression.

$x n + m n - (x + n) m = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.4

Apply the distributive property.

$x n + m n + (- x - n) m = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.1.5

Apply the distributive property.

$x n + m n - x m - n m = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.2

Combine the opposite terms in $x n + m n - x m - n m$ .

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Step 2.2.2.1

Reorder the factors in the terms $m n$ and $- n m$ .

$x n + m n - x m - m n = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.2.2

Subtract $m n$ from $m n$ .

$x n - x m + 0 = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.2.2.3

Add $x n - x m$ and $0$ .

$x n - x m = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $m n$ .

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Step 2.3.1.1

Cancel the common factor.

$x n - x m = \frac{m^{2} + n^{2}}{m n} (m n) - 2 (m n)$

Step 2.3.1.2

Rewrite the expression.

$x n - x m = m^{2} + n^{2} - 2 (m n)$

$x n - x m = m^{2} + n^{2} - 2 m n$

Step 3

Solve the equation.

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Step 3.1

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

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Step 3.1.1

Factor $x$ out of $x n$ .

$x (n) - x m = m^{2} + n^{2} - 2 m n$

Step 3.1.2

Factor $x$ out of $- x m$ .

$x (n) + x (- 1 m) = m^{2} + n^{2} - 2 m n$

Step 3.1.3

Factor $x$ out of $x (n) + x (- 1 m)$ .

$x (n - 1 m) = m^{2} + n^{2} - 2 m n$

Step 3.2

Rewrite $- 1 m$ as $- m$ .

$x (n - m) = m^{2} + n^{2} - 2 m n$

Step 3.3

Divide each term in $x (n - m) = m^{2} + n^{2} - 2 m n$ by $n - m$ and simplify.

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Step 3.3.1

Divide each term in $x (n - m) = m^{2} + n^{2} - 2 m n$ by $n - m$ .

$\frac{x (n - m)}{n - m} = \frac{m^{2}}{n - m} + \frac{n^{2}}{n - m} + \frac{- 2 m n}{n - m}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $n - m$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{x (n - m)}{n - m} = \frac{m^{2}}{n - m} + \frac{n^{2}}{n - m} + \frac{- 2 m n}{n - m}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{m^{2}}{n - m} + \frac{n^{2}}{n - m} + \frac{- 2 m n}{n - m}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Combine into one fraction.

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Step 3.3.3.1.1

Move the negative in front of the fraction.

$x = \frac{m^{2}}{n - m} + \frac{n^{2}}{n - m} - \frac{2 m n}{n - m}$

Step 3.3.3.1.2

Combine the numerators over the common denominator.

$x = \frac{m^{2} + n^{2}}{n - m} - \frac{2 m n}{n - m}$

Step 3.3.3.1.3

Combine the numerators over the common denominator.

$x = \frac{m^{2} + n^{2} - 2 m n}{n - m}$

Step 3.3.3.2

Factor using the perfect square rule.

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Step 3.3.3.2.1

Rearrange terms.

$x = \frac{m^{2} - 2 m n + n^{2}}{n - m}$

Step 3.3.3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 m n = 2 \cdot m \cdot n$

Step 3.3.3.2.3

Rewrite the polynomial.

$x = \frac{m^{2} - 2 \cdot m \cdot n + n^{2}}{n - m}$

Step 3.3.3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = m$ and $b = n$ .

$x = \frac{{(m - n)}^{2}}{n - m}$

Step 3.3.3.3

Cancel the common factor of ${(m - n)}^{2}$ and $n - m$ .

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Step 3.3.3.3.1

Factor $- 1$ out of $m$ .

$x = \frac{{(- 1 (- m) - n)}^{2}}{n - m}$

Step 3.3.3.3.2

Factor $- 1$ out of $- n$ .

$x = \frac{{(- 1 (- m) - (n))}^{2}}{n - m}$

Step 3.3.3.3.3

Factor $- 1$ out of $- 1 (- m) - (n)$ .

$x = \frac{{(- 1 (- m + n))}^{2}}{n - m}$

Step 3.3.3.3.4

Apply the product rule to $- 1 (- m + n)$ .

$x = \frac{{(- 1)}^{2} {(- m + n)}^{2}}{n - m}$

Step 3.3.3.3.5

Raise $- 1$ to the power of $2$ .

$x = \frac{1 {(- m + n)}^{2}}{n - m}$

Step 3.3.3.3.6

Multiply ${(- m + n)}^{2}$ by $1$ .

$x = \frac{{(- m + n)}^{2}}{n - m}$

Step 3.3.3.3.7

Reorder terms.

$x = \frac{{(- m + n)}^{2}}{- m + n}$

Step 3.3.3.3.8

Factor $- m + n$ out of ${(- m + n)}^{2}$ .

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

Step 3.3.3.3.9

Cancel the common factors.

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Step 3.3.3.3.9.1

Multiply by $1$ .

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

Step 3.3.3.3.9.2

Cancel the common factor.

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

Step 3.3.3.3.9.3

Rewrite the expression.

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

Step 3.3.3.3.9.4

Divide $- m + n$ by $1$ .

$x = - m + n$