Solve varepsilon=pi/E(sigma_{1}-v(sigma_{2}+sigma_{3}))

$$\varepsilon = \frac { \pi } { E } ( \sigma _ { 1 } - v ( \sigma _ { 2 } + \sigma _ { 3 } ) )$$

Solve for E

$$\left\{\begin{matrix}E=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }\text{, }&\sigma _{1}\neq v\left(\sigma _{2}+\sigma _{3}\right)\text{ and }\epsilon \neq 0\text{ and }\sigma _{1}\neq v\sigma _{2}+v\sigma _{3}\\E\neq 0\text{, }&\epsilon =0\text{ and }\sigma _{1}=v\left(\sigma _{2}+\sigma _{3}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable E cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by E.

$$\epsilon E=\pi \left(\sigma _{1}-v\left(\sigma _{2}+\sigma _{3}\right)\right)$$

Use the distributive property to multiply v by \sigma _{2}+\sigma _{3}.

$$\epsilon E=\pi \left(\sigma _{1}-\left(v\sigma _{2}+v\sigma _{3}\right)\right)$$

To find the opposite of v\sigma _{2}+v\sigma _{3}, find the opposite of each term.

$$\epsilon E=\pi \left(\sigma _{1}-v\sigma _{2}-v\sigma _{3}\right)$$

Use the distributive property to multiply \pi by \sigma _{1}-v\sigma _{2}-v\sigma _{3}.

$$\epsilon E=\pi \sigma _{1}-\pi v\sigma _{2}-\pi v\sigma _{3}$$

The equation is in standard form.

$$\epsilon E=\pi \sigma _{1}-\pi v\sigma _{3}-\pi v\sigma _{2}$$

Divide both sides by \epsilon .

$$\frac{\epsilon E}{\epsilon }=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }$$

Dividing by \epsilon undoes the multiplication by \epsilon .

$$E=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }$$

Variable E cannot be equal to 0.

$$E=\frac{\pi \left(\sigma _{1}-v\sigma _{3}-v\sigma _{2}\right)}{\epsilon }\text{, }E\neq 0$$

Solve for v

$$\left\{\begin{matrix}v=\frac{\pi \sigma _{1}-E\epsilon }{\pi \left(\sigma _{2}+\sigma _{3}\right)}\text{, }&E\neq 0\text{ and }\sigma _{2}\neq -\sigma _{3}\\v\in \mathrm{R}\text{, }&\sigma _{1}=\frac{E\epsilon }{\pi }\text{ and }\sigma _{2}=-\sigma _{3}\text{ and }E\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by E.

$$\epsilon E=\pi \left(\sigma _{1}-v\left(\sigma _{2}+\sigma _{3}\right)\right)$$

Use the distributive property to multiply v by \sigma _{2}+\sigma _{3}.

$$\epsilon E=\pi \left(\sigma _{1}-\left(v\sigma _{2}+v\sigma _{3}\right)\right)$$

To find the opposite of v\sigma _{2}+v\sigma _{3}, find the opposite of each term.

$$\epsilon E=\pi \left(\sigma _{1}-v\sigma _{2}-v\sigma _{3}\right)$$

Use the distributive property to multiply \pi by \sigma _{1}-v\sigma _{2}-v\sigma _{3}.

$$\epsilon E=\pi \sigma _{1}-\pi v\sigma _{2}-\pi v\sigma _{3}$$

Swap sides so that all variable terms are on the left hand side.

$$\pi \sigma _{1}-\pi v\sigma _{2}-\pi v\sigma _{3}=\epsilon E$$

Subtract \pi \sigma _{1} from both sides.

$$-\pi v\sigma _{2}-\pi v\sigma _{3}=\epsilon E-\pi \sigma _{1}$$

Reorder the terms.

$$-\pi v\sigma _{2}-\pi v\sigma _{3}=E\epsilon -\pi \sigma _{1}$$

Combine all terms containing v.

$$\left(-\pi \sigma _{2}-\pi \sigma _{3}\right)v=E\epsilon -\pi \sigma _{1}$$

Divide both sides by -\pi \sigma _{2}-\pi \sigma _{3}.

$$\frac{\left(-\pi \sigma _{2}-\pi \sigma _{3}\right)v}{-\pi \sigma _{2}-\pi \sigma _{3}}=\frac{E\epsilon -\pi \sigma _{1}}{-\pi \sigma _{2}-\pi \sigma _{3}}$$

Dividing by -\pi \sigma _{2}-\pi \sigma _{3} undoes the multiplication by -\pi \sigma _{2}-\pi \sigma _{3}.

$$v=\frac{E\epsilon -\pi \sigma _{1}}{-\pi \sigma _{2}-\pi \sigma _{3}}$$

Divide \epsilon E-\pi \sigma _{1} by -\pi \sigma _{2}-\pi \sigma _{3}.

$$v=\frac{E\epsilon -\pi \sigma _{1}}{-\pi \left(\sigma _{2}+\sigma _{3}\right)}$$