Solve 3x(5/12t^2-7/6t-8)=t+4

$$3 x ( \frac { 5 } { 12 } t ^ { 2 } - \frac { 7 } { 6 } t - 8 ) = t + 4$$

Solve for x

$$x=\frac{4\left(t+4\right)}{\left(t-6\right)\left(5t+16\right)}$$

$$t\neq -\frac{16}{5}\text{ and }t\neq 6$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3x by \frac{5}{12}t^{2}-\frac{7}{6}t-8.

$$\frac{5}{4}xt^{2}-\frac{7}{2}xt-24x=t+4$$

Combine all terms containing x.

$$\left(\frac{5}{4}t^{2}-\frac{7}{2}t-24\right)x=t+4$$

The equation is in standard form.

$$\left(\frac{5t^{2}}{4}-\frac{7t}{2}-24\right)x=t+4$$

Divide both sides by \frac{5}{4}t^{2}-\frac{7}{2}t-24.

$$\frac{\left(\frac{5t^{2}}{4}-\frac{7t}{2}-24\right)x}{\frac{5t^{2}}{4}-\frac{7t}{2}-24}=\frac{t+4}{\frac{5t^{2}}{4}-\frac{7t}{2}-24}$$

Dividing by \frac{5}{4}t^{2}-\frac{7}{2}t-24 undoes the multiplication by \frac{5}{4}t^{2}-\frac{7}{2}t-24.

$$x=\frac{t+4}{\frac{5t^{2}}{4}-\frac{7t}{2}-24}$$

Divide t+4 by \frac{5}{4}t^{2}-\frac{7}{2}t-24.

$$x=\frac{4\left(t+4\right)}{\left(t-6\right)\left(5t+16\right)}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{\sqrt{529x^{2}+108x+4}+7x+2}{5x}\text{; }t=\frac{-\sqrt{529x^{2}+108x+4}+7x+2}{5x}\text{, }&x\neq 0\\t=-4\text{, }&x=0\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=\frac{\sqrt{529x^{2}+108x+4}+7x+2}{5x}\text{; }t=\frac{-\sqrt{529x^{2}+108x+4}+7x+2}{5x}\text{, }&x\leq \frac{-20\sqrt{2}-54}{529}\text{ or }\left(x\neq 0\text{ and }x\geq \frac{20\sqrt{2}-54}{529}\right)\\t=-4\text{, }&x=0\end{matrix}\right.$$