Solve for K l=(K+1)/k-(m+1)/m

$l = \frac{K + 1}{k} - \frac{m + 1}{m}$

Step 1

Rewrite the equation as $\frac{K + 1}{k} - \frac{m + 1}{m} = l$ .

$\frac{K + 1}{k} - \frac{m + 1}{m} = l$

Step 2

Simplify $\frac{K + 1}{k} - \frac{m + 1}{m}$ .

To write $\frac{K + 1}{k}$ as a fraction with a common denominator, multiply by $\frac{m}{m}$ .

$\frac{K + 1}{k} \cdot \frac{m}{m} - \frac{m + 1}{m} = l$

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

$\frac{K + 1}{k} \cdot \frac{m}{m} - \frac{m + 1}{m} \cdot \frac{k}{k} = l$

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

Multiply $\frac{K + 1}{k}$ by $\frac{m}{m}$ .

$\frac{(K + 1) m}{k m} - \frac{m + 1}{m} \cdot \frac{k}{k} = l$

Multiply $\frac{m + 1}{m}$ by $\frac{k}{k}$ .

$\frac{(K + 1) m}{k m} - \frac{(m + 1) k}{m k} = l$

Reorder the factors of $m k$ .

$\frac{(K + 1) m}{k m} - \frac{(m + 1) k}{k m} = l$

Combine the numerators over the common denominator.

$\frac{(K + 1) m - (m + 1) k}{k m} = l$

Simplify the numerator.

Apply the distributive property.

$\frac{K m + 1 m - (m + 1) k}{k m} = l$

Multiply $m$ by $1$ .

$\frac{K m + m - (m + 1) k}{k m} = l$

Apply the distributive property.

$\frac{K m + m + (- m - 1 \cdot 1) k}{k m} = l$

Multiply $- 1$ by $1$ .

$\frac{K m + m + (- m - 1) k}{k m} = l$

Apply the distributive property.

$\frac{K m + m - m k - 1 k}{k m} = l$

Rewrite $- 1 k$ as $- k$ .

$\frac{K m + m - m k - k}{k m} = l$

Step 3

Multiply both sides by $k m$ .

$\frac{K m + m - m k - k}{k m} (k m) = l (k m)$

Step 4

Simplify.

Simplify the left side.

Simplify $\frac{K m + m - m k - k}{k m} (k m)$ .

Cancel the common factor of $k m$ .

Cancel the common factor.

$\frac{K m + m - m k - k}{k m} (k m) = l (k m)$

Rewrite the expression.

$K m + m - m k - k = l (k m)$

Simplify the expression.

Reorder $K$ and $m$ .

$m K + m - m k - k = l (k m)$

Move $m$ .

$m K + m - 1 k m - k = l (k m)$

Move $m$ .

$m K - 1 k m - k + m = l (k m)$

Simplify the right side.

Remove parentheses.

$m K - 1 k m - k + m = l k m$

Step 5

Solve for $K$ .

Rewrite $- 1 k$ as $- k$ .

$m K - k m - k + m = l k m$

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

Add $k m$ to both sides of the equation.

$m K - k + m = l k m + k m$

Add $k$ to both sides of the equation.

$m K + m = l k m + k m + k$

Subtract $m$ from both sides of the equation.

$m K = l k m + k m + k - m$

Divide each term in $m K = l k m + k m + k - m$ by $m$ and simplify.

Divide each term in $m K = l k m + k m + k - m$ by $m$ .

$\frac{m K}{m} = \frac{l k m}{m} + \frac{k m}{m} + \frac{k}{m} + \frac{- m}{m}$

Simplify the left side.

Cancel the common factor of $m$ .

Cancel the common factor.

$\frac{m K}{m} = \frac{l k m}{m} + \frac{k m}{m} + \frac{k}{m} + \frac{- m}{m}$

Divide $K$ by $1$ .

$K = \frac{l k m}{m} + \frac{k m}{m} + \frac{k}{m} + \frac{- m}{m}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $m$ .

Cancel the common factor.

$K = \frac{l k m}{m} + \frac{k m}{m} + \frac{k}{m} + \frac{- m}{m}$

Divide $l k$ by $1$ .

$K = l k + \frac{k m}{m} + \frac{k}{m} + \frac{- m}{m}$

Cancel the common factor of $m$ .

Cancel the common factor.

$K = l k + \frac{k m}{m} + \frac{k}{m} + \frac{- m}{m}$

Divide $k$ by $1$ .

$K = l k + k + \frac{k}{m} + \frac{- m}{m}$

Cancel the common factor of $m$ .

Cancel the common factor.

$K = l k + k + \frac{k}{m} + \frac{- m}{m}$

Divide $- 1$ by $1$ .

$K = l k + k + \frac{k}{m} - 1$

Solve for K l=(K+1)/k-(m+1)/m

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