Solve y-1=(y+2)1/4e^6x

$$y - 1 = ( y + 2 ) \frac { 1 } { 4 } e ^ { 6 x }$$

Solve for y

$$y=\frac{2\left(e^{6x}+2\right)}{4-e^{6x}}$$

$$x\neq \frac{\ln(2)}{3}$$

Steps for Solving Linear Equation

Use the distributive property to multiply y+2 by \frac{1}{4}.

$$y-1=\left(\frac{1}{4}y+\frac{1}{2}\right)e^{6x}$$

Use the distributive property to multiply \frac{1}{4}y+\frac{1}{2} by e^{6x}.

$$y-1=\frac{1}{4}ye^{6x}+\frac{1}{2}e^{6x}$$

Subtract \frac{1}{4}ye^{6x} from both sides.

$$y-1-\frac{1}{4}ye^{6x}=\frac{1}{2}e^{6x}$$

Add 1 to both sides.

$$y-\frac{1}{4}ye^{6x}=\frac{1}{2}e^{6x}+1$$

Combine all terms containing y.

$$\left(1-\frac{1}{4}e^{6x}\right)y=\frac{1}{2}e^{6x}+1$$

The equation is in standard form.

$$\left(-\frac{e^{6x}}{4}+1\right)y=\frac{e^{6x}}{2}+1$$

Divide both sides by 1-\frac{1}{4}e^{6x}.

$$\frac{\left(-\frac{e^{6x}}{4}+1\right)y}{-\frac{e^{6x}}{4}+1}=\frac{\frac{e^{6x}}{2}+1}{-\frac{e^{6x}}{4}+1}$$

Dividing by 1-\frac{1}{4}e^{6x} undoes the multiplication by 1-\frac{1}{4}e^{6x}.

$$y=\frac{\frac{e^{6x}}{2}+1}{-\frac{e^{6x}}{4}+1}$$

Divide \frac{e^{6x}}{2}+1 by 1-\frac{1}{4}e^{6x}.

$$y=\frac{2\left(e^{6x}+2\right)}{4-e^{6x}}$$

Solve for x

$$x=\frac{\ln(\frac{y-1}{y+2})+2\ln(2)}{6}$$

$$y<-2\text{ or }y>1$$