Solve 4sinx=3-a+1/a-1

$$4 \sin x = 3 - \frac { a + 1 } { a - 1 }$$

Solve for a

$$a=-\frac{2\left(-\sin(x)+1\right)}{2\sin(x)-1}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\left(x=2\pi n_{1}+\frac{5\pi }{6}\text{ or }x=2\pi n_{1}+\frac{\pi }{6}\right)$$

Steps for Solving Linear Equation

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by a-1.

$$4\sin(x)\left(a-1\right)=\left(a-1\right)\times 3-\left(a+1\right)$$

Use the distributive property to multiply 4\sin(x) by a-1.

$$4\sin(x)a-4\sin(x)=\left(a-1\right)\times 3-\left(a+1\right)$$

Use the distributive property to multiply a-1 by 3.

$$4\sin(x)a-4\sin(x)=3a-3-\left(a+1\right)$$

To find the opposite of a+1, find the opposite of each term.

$$4\sin(x)a-4\sin(x)=3a-3-a-1$$

Combine 3a and -a to get 2a.

$$4\sin(x)a-4\sin(x)=2a-3-1$$

Subtract 1 from -3 to get -4.

$$4\sin(x)a-4\sin(x)=2a-4$$

Subtract 2a from both sides.

$$4\sin(x)a-4\sin(x)-2a=-4$$

Add 4\sin(x) to both sides.

$$4\sin(x)a-2a=-4+4\sin(x)$$

Combine all terms containing a.

$$\left(4\sin(x)-2\right)a=-4+4\sin(x)$$

The equation is in standard form.

$$\left(4\sin(x)-2\right)a=4\sin(x)-4$$

Divide both sides by 4\sin(x)-2.

$$\frac{\left(4\sin(x)-2\right)a}{4\sin(x)-2}=\frac{4\left(\sin(x)-1\right)}{4\sin(x)-2}$$

Dividing by 4\sin(x)-2 undoes the multiplication by 4\sin(x)-2.

$$a=\frac{4\left(\sin(x)-1\right)}{4\sin(x)-2}$$

Divide 4\left(-1+\sin(x)\right) by 4\sin(x)-2.

$$a=\frac{2\left(\sin(x)-1\right)}{2\sin(x)-1}$$

Variable a cannot be equal to 1.

$$a=\frac{2\left(\sin(x)-1\right)}{2\sin(x)-1}\text{, }a\neq 1$$