Solve for a (a+b/c)(d-e)=f

$(a + \frac{b}{c}) (d - e) = f$

Step 1

Divide each term in $(a + \frac{b}{c}) (d - e) = f$ by $d - e$ and simplify.

Divide each term in $(a + \frac{b}{c}) (d - e) = f$ by $d - e$ .

$\frac{(a + \frac{b}{c}) (d - e)}{d - e} = \frac{f}{d - e}$

Simplify the left side.

Cancel the common factor of $d - e$ .

Cancel the common factor.

$\frac{(a + \frac{b}{c}) (d - e)}{d - e} = \frac{f}{d - e}$

Divide $a + \frac{b}{c}$ by $1$ .

$a + \frac{b}{c} = \frac{f}{d - e}$

Step 2

Move all terms containing variables to the left side of the equation.

Subtract $\frac{f}{d - e}$ from both sides of the equation.

$a + \frac{b}{c} - \frac{f}{d - e} = 0$

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

$\frac{a (d - e)}{d - e} - \frac{f}{d - e} + \frac{b}{c} = 0$

Combine the numerators over the common denominator.

$\frac{a (d - e) - f}{d - e} + \frac{b}{c} = 0$

Apply the distributive property.

$\frac{a d + a (- e) - f}{d - e} + \frac{b}{c} = 0$

To write $\frac{a d + a (- e) - f}{d - e}$ as a fraction with a common denominator, multiply by $\frac{c}{c}$ .

$\frac{a d + a (- e) - f}{d - e} \cdot \frac{c}{c} + \frac{b}{c} = 0$

To write $\frac{b}{c}$ as a fraction with a common denominator, multiply by $\frac{d - e}{d - e}$ .

$\frac{a d + a (- e) - f}{d - e} \cdot \frac{c}{c} + \frac{b}{c} \cdot \frac{d - e}{d - e} = 0$

Write each expression with a common denominator of $(d - e) c$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a d + a (- e) - f}{d - e}$ by $\frac{c}{c}$ .

$\frac{(a d + a (- e) - f) c}{(d - e) c} + \frac{b}{c} \cdot \frac{d - e}{d - e} = 0$

Multiply $\frac{b}{c}$ by $\frac{d - e}{d - e}$ .

$\frac{(a d + a (- e) - f) c}{(d - e) c} + \frac{b (d - e)}{c (d - e)} = 0$

Reorder the factors of $(d - e) c$ .

$\frac{(a d + a (- e) - f) c}{c (d - e)} + \frac{b (d - e)}{c (d - e)} = 0$

Combine the numerators over the common denominator.

$\frac{(a d + a (- e) - f) c + b (d - e)}{c (d - e)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{a d c + a (- e) c - f c + b (d - e)}{c (d - e)} = 0$

Apply the distributive property.

$\frac{a d c + a (- e) c - f c + b d + b (- e)}{c (d - e)} = 0$

Reorder factors in $\frac{a d c + a (- e) c - f c + b d + b (- e)}{c (d - e)}$ .

$\frac{a d c - a e c - f c + b d - b e}{c (d - e)} = 0$

Step 3

Set the numerator equal to zero.

$a d c - a e c - f c + b d - b e = 0$

Step 4

Solve the equation for $a$ .

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

Add $f c$ to both sides of the equation.

$a d c - a e c + b d - b e = f c$

Subtract $b d$ from both sides of the equation.

$a d c - a e c - b e = f c - b d$

Add $b e$ to both sides of the equation.

$a d c - a e c = f c - b d + b e$

Factor $a c$ out of $a d c - a e c$ .

Factor $a c$ out of $a d c$ .

$a c (d) - a e c = f c - b d + b e$

Factor $a c$ out of $- a e c$ .

$a c (d) + a c (- 1 e) = f c - b d + b e$

Factor $a c$ out of $a c (d) + a c (- 1 e)$ .

$a c (d - 1 e) = f c - b d + b e$

Rewrite $- 1 e$ as $- e$ .

$a c (d - e) = f c - b d + b e$

Divide each term in $a c (d - e) = f c - b d + b e$ by $c (d - e)$ and simplify.

Divide each term in $a c (d - e) = f c - b d + b e$ by $c (d - e)$ .

$\frac{a c (d - e)}{c (d - e)} = \frac{f c}{c (d - e)} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Simplify the left side.

Cancel the common factor of $c$ .

Cancel the common factor.

$\frac{a c (d - e)}{c (d - e)} = \frac{f c}{c (d - e)} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Rewrite the expression.

$\frac{a (d - e)}{d - e} = \frac{f c}{c (d - e)} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Cancel the common factor of $d - e$ .

Cancel the common factor.

$\frac{a (d - e)}{d - e} = \frac{f c}{c (d - e)} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Divide $a$ by $1$ .

$a = \frac{f c}{c (d - e)} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $c$ .

Cancel the common factor.

$a = \frac{f c}{c (d - e)} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Rewrite the expression.

$a = \frac{f}{d - e} + \frac{- b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Move the negative in front of the fraction.

$a = \frac{f}{d - e} - \frac{b d}{c (d - e)} + \frac{b e}{c (d - e)}$

To write $\frac{f}{d - e}$ as a fraction with a common denominator, multiply by $\frac{c}{c}$ .

$a = \frac{f}{d - e} \cdot \frac{c}{c} - \frac{b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Write each expression with a common denominator of $(d - e) c$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{f}{d - e}$ by $\frac{c}{c}$ .

$a = \frac{f c}{(d - e) c} - \frac{b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Reorder the factors of $(d - e) c$ .

$a = \frac{f c}{c (d - e)} - \frac{b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Combine the numerators over the common denominator.

$a = \frac{f c - b d}{c (d - e)} + \frac{b e}{c (d - e)}$

Combine the numerators over the common denominator.

$a = \frac{f c - b d + b e}{c (d - e)}$

Solve for a (a+b/c)(d-e)=f

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