Solve frac{{c}_{2}rleft(k-rright)}{k}T-frac{{c}_{1}}{T^2}=0

$$\frac{ { c }_{ 2 } r \left( k-r \right) }{ k } T- \frac{ { c }_{ 1 } }{ { T }^{ 2 } } =0$$

Solve for c_1

$$c_{1}=\frac{c_{2}r\left(k-r\right)T^{3}}{k}$$

$$k\neq 0\text{ and }T\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by kT^{2}, the least common multiple of k,T^{2}.

$$T^{2}c_{2}r\left(k-r\right)T-kc_{1}=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$T^{3}c_{2}r\left(k-r\right)-kc_{1}=0$$

Use the distributive property to multiply T^{3}c_{2}r by k-r.

$$T^{3}c_{2}rk-T^{3}c_{2}r^{2}-kc_{1}=0$$

Subtract T^{3}c_{2}rk from both sides. Anything subtracted from zero gives its negation.

$$-T^{3}c_{2}r^{2}-kc_{1}=-T^{3}c_{2}rk$$

Add T^{3}c_{2}r^{2} to both sides.

$$-kc_{1}=-T^{3}c_{2}rk+T^{3}c_{2}r^{2}$$

Reorder the terms.

$$-c_{1}k=c_{2}r^{2}T^{3}-c_{2}krT^{3}$$

The equation is in standard form.

$$\left(-k\right)c_{1}=c_{2}r^{2}T^{3}-c_{2}krT^{3}$$

Divide both sides by -k.

$$\frac{\left(-k\right)c_{1}}{-k}=\frac{c_{2}r\left(r-k\right)T^{3}}{-k}$$

Dividing by -k undoes the multiplication by -k.

$$c_{1}=\frac{c_{2}r\left(r-k\right)T^{3}}{-k}$$

Divide c_{2}r\left(r-k\right)T^{3} by -k.

$$c_{1}=-\frac{c_{2}r\left(r-k\right)T^{3}}{k}$$

Solve for T

$$\left\{\begin{matrix}T=\sqrt[3]{\frac{c_{1}k}{c_{2}r\left(k-r\right)}}\text{, }&k\neq 0\text{ and }c_{1}\neq 0\text{ and }r\neq k\text{ and }r\neq 0\text{ and }c_{2}\neq 0\\T\neq 0\text{, }&\left(r=k\text{ or }r=0\text{ or }c_{2}=0\right)\text{ and }c_{1}=0\text{ and }k\neq 0\end{matrix}\right.$$