Algebra Examples

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

Step 1

Multiply both sides by $n$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{x - m}{n} n$ .

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Step 2.1.1.1

Cancel the common factor of $n$ .

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Step 2.1.1.1.1

Cancel the common factor.

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

Step 2.1.1.1.2

Rewrite the expression.

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

Step 2.1.1.2

Reorder $x$ and $- m$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(2 - \frac{x - n}{m}) n$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Combine the numerators over the common denominator.

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

Step 2.2.1.3

Simplify the numerator.

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Step 2.2.1.3.1

Apply the distributive property.

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

Step 2.2.1.3.2

Multiply $- - n$ .

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Step 2.2.1.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.1.3.2.2

Multiply $n$ by $1$ .

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

Step 2.2.1.4

Combine fractions.

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Step 2.2.1.4.1

Combine $\frac{2 m - x + n}{m}$ and $n$ .

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

Step 2.2.1.4.2

Reorder factors in $\frac{(2 m - x + n) n}{m}$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $m$ .

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

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

Simplify $(- m + x) m$ .

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Step 3.2.1.1.1

Apply the distributive property.

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

Step 3.2.1.1.2

Multiply $m$ by $m$ by adding the exponents.

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Step 3.2.1.1.2.1

Move $m$ .

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

Step 3.2.1.1.2.2

Multiply $m$ by $m$ .

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

Step 3.2.1.1.3

Reorder $x$ and $m$ .

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

Step 3.2.1.1.4

Reorder $- m^{2}$ and $m x$ .

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{n (2 m - x + n)}{m} m$ .

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Step 3.2.2.1.1

Cancel the common factor of $m$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

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

Step 3.2.2.1.1.2

Rewrite the expression.

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.2.1.3.3

Multiply $n$ by $n$ .

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

Step 3.2.2.1.4

Move $n$ .

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

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Add $m^{2}$ to both sides of the equation.

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

Step 3.3.3

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

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Step 3.3.3.1

Factor $x$ out of $m x$ .

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

Step 3.3.3.2

Factor $x$ out of $n x$ .

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

Step 3.3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Cancel the common factor of $m + n$ .

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Step 3.3.4.2.1.1

Cancel the common factor.

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

Step 3.3.4.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Combine into one fraction.

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Step 3.3.4.3.1.1

Combine the numerators over the common denominator.

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

Step 3.3.4.3.1.2

Combine the numerators over the common denominator.

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

Step 3.3.4.3.2

Factor using the perfect square rule.

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Step 3.3.4.3.2.1

Rearrange terms.

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

Step 3.3.4.3.2.2

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

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

Step 3.3.4.3.2.3

Rewrite the polynomial.

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

Step 3.3.4.3.2.4

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

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

Step 3.3.4.3.3

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

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Step 3.3.4.3.3.1

Reorder terms.

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

Step 3.3.4.3.3.2

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

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

Step 3.3.4.3.3.3

Cancel the common factors.

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Step 3.3.4.3.3.3.1

Multiply by $1$ .

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

Step 3.3.4.3.3.3.2

Cancel the common factor.

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

Step 3.3.4.3.3.3.3

Rewrite the expression.

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

Step 3.3.4.3.3.3.4

Divide $m + n$ by $1$ .

$x = m + n$