Solve varepsilonA/d_{1}+varepsilonA/d_{2}=frac{(d_{1}fd_{2})varepsilonA}{d_{1}d_{2}}

$$\frac { \varepsilon A } { d _ { 1 } } + \frac { \varepsilon A } { d _ { 2 } } = \frac { ( d _ { 1 } f d _ { 2 } ) \varepsilon A } { d _ { 1 } d _ { 2 } }$$

Solve for A

$$\left\{\begin{matrix}A=0\text{, }&d_{1}\neq 0\text{ and }d_{2}\neq 0\\A\in \mathrm{R}\text{, }&\left(d_{1}=-\frac{d_{2}}{1-d_{2}f}\text{ and }d_{2}\neq \frac{1}{f}\text{ and }f\neq 0\text{ and }d_{2}\neq 0\right)\text{ or }\left(d_{1}=-d_{2}\text{ and }f=0\text{ and }d_{2}\neq 0\right)\text{ or }\left(\epsilon =0\text{ and }d_{1}\neq 0\text{ and }d_{2}\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by d_{1}d_{2}, the least common multiple of d_{1},d_{2},d_{1}d_{2}.

$$d_{2}\epsilon A+d_{1}\epsilon A=d_{1}fd_{2}\epsilon A$$

Subtract d_{1}fd_{2}\epsilon A from both sides.

$$d_{2}\epsilon A+d_{1}\epsilon A-d_{1}fd_{2}\epsilon A=0$$

Reorder the terms.

$$-Ad_{1}d_{2}f\epsilon +Ad_{1}\epsilon +Ad_{2}\epsilon =0$$

Combine all terms containing A.

$$\left(-d_{1}d_{2}f\epsilon +d_{1}\epsilon +d_{2}\epsilon \right)A=0$$

The equation is in standard form.

$$\left(d_{2}\epsilon +d_{1}\epsilon -d_{1}d_{2}f\epsilon \right)A=0$$

Divide 0 by -d_{1}d_{2}f\epsilon +d_{1}\epsilon +d_{2}\epsilon .

$$A=0$$

Solve for d_1

$$\left\{\begin{matrix}d_{1}=-\frac{d_{2}}{1-d_{2}f}\text{, }&\left(f=0\text{ or }d_{2}\neq \frac{1}{f}\right)\text{ and }d_{2}\neq 0\\d_{1}\neq 0\text{, }&\left(A=0\text{ or }\epsilon =0\right)\text{ and }d_{2}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable d_{1} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by d_{1}d_{2}, the least common multiple of d_{1},d_{2},d_{1}d_{2}.

$$d_{2}\epsilon A+d_{1}\epsilon A=d_{1}fd_{2}\epsilon A$$

Subtract d_{1}fd_{2}\epsilon A from both sides.

$$d_{2}\epsilon A+d_{1}\epsilon A-d_{1}fd_{2}\epsilon A=0$$

Subtract d_{2}\epsilon A from both sides. Anything subtracted from zero gives its negation.

$$d_{1}\epsilon A-d_{1}fd_{2}\epsilon A=-d_{2}\epsilon A$$

Combine all terms containing d_{1}.

$$\left(\epsilon A-fd_{2}\epsilon A\right)d_{1}=-d_{2}\epsilon A$$

The equation is in standard form.

$$\left(A\epsilon -Ad_{2}f\epsilon \right)d_{1}=-Ad_{2}\epsilon$$

Divide both sides by \epsilon A-fd_{2}\epsilon A.

$$\frac{\left(A\epsilon -Ad_{2}f\epsilon \right)d_{1}}{A\epsilon -Ad_{2}f\epsilon }=-\frac{Ad_{2}\epsilon }{A\epsilon -Ad_{2}f\epsilon }$$

Dividing by \epsilon A-fd_{2}\epsilon A undoes the multiplication by \epsilon A-fd_{2}\epsilon A.

$$d_{1}=-\frac{Ad_{2}\epsilon }{A\epsilon -Ad_{2}f\epsilon }$$

Divide -d_{2}\epsilon A by \epsilon A-fd_{2}\epsilon A.

$$d_{1}=-\frac{d_{2}}{1-d_{2}f}$$

Variable d_{1} cannot be equal to 0.

$$d_{1}=-\frac{d_{2}}{1-d_{2}f}\text{, }d_{1}\neq 0$$