Solve x=1/2-1{2-frac{1/2-lambda}}

$$x = \frac { 1 } { 2 - \frac { 1 } { 2 - \frac { 1 } { 2 - \lambda } } }$$

Solve for λ

$$\lambda =-\frac{4x-3}{2-3x}$$

$$x\neq \frac{2}{3}\text{ and }x\neq 0\text{ and }x\neq \frac{1}{2}$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{2-\lambda }{2-\lambda }.

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

Since \frac{2\left(2-\lambda \right)}{2-\lambda } and \frac{1}{2-\lambda } have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(2-\lambda \right)-1.

$$x=\frac{1}{2-\frac{1}{\frac{4-2\lambda -1}{2-\lambda }}}$$

Combine like terms in 4-2\lambda -1.

$$x=\frac{1}{2-\frac{1}{\frac{3-2\lambda }{2-\lambda }}}$$

Variable \lambda cannot be equal to 2 since division by zero is not defined. Divide 1 by \frac{3-2\lambda }{2-\lambda } by multiplying 1 by the reciprocal of \frac{3-2\lambda }{2-\lambda }.

$$x=\frac{1}{2-\frac{2-\lambda }{3-2\lambda }}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{3-2\lambda }{3-2\lambda }.

$$x=\frac{1}{\frac{2\left(3-2\lambda \right)}{3-2\lambda }-\frac{2-\lambda }{3-2\lambda }}$$

Since \frac{2\left(3-2\lambda \right)}{3-2\lambda } and \frac{2-\lambda }{3-2\lambda } have the same denominator, subtract them by subtracting their numerators.

$$x=\frac{1}{\frac{2\left(3-2\lambda \right)-\left(2-\lambda \right)}{3-2\lambda }}$$

Do the multiplications in 2\left(3-2\lambda \right)-\left(2-\lambda \right).

$$x=\frac{1}{\frac{6-4\lambda -2+\lambda }{3-2\lambda }}$$

Combine like terms in 6-4\lambda -2+\lambda .

$$x=\frac{1}{\frac{4-3\lambda }{3-2\lambda }}$$

Variable \lambda cannot be equal to \frac{3}{2} since division by zero is not defined. Divide 1 by \frac{4-3\lambda }{3-2\lambda } by multiplying 1 by the reciprocal of \frac{4-3\lambda }{3-2\lambda }.

$$x=\frac{3-2\lambda }{4-3\lambda }$$

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

$$\frac{3-2\lambda }{4-3\lambda }=x$$

Variable \lambda cannot be equal to \frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by -3\lambda +4.

$$3-2\lambda =x\left(-3\lambda +4\right)$$

Use the distributive property to multiply x by -3\lambda +4.

$$3-2\lambda =-3x\lambda +4x$$

Add 3x\lambda to both sides.

$$3-2\lambda +3x\lambda =4x$$

Subtract 3 from both sides.

$$-2\lambda +3x\lambda =4x-3$$

Combine all terms containing \lambda .

$$\left(-2+3x\right)\lambda =4x-3$$

The equation is in standard form.

$$\left(3x-2\right)\lambda =4x-3$$

Divide both sides by -2+3x.

$$\frac{\left(3x-2\right)\lambda }{3x-2}=\frac{4x-3}{3x-2}$$

Dividing by -2+3x undoes the multiplication by -2+3x.

$$\lambda =\frac{4x-3}{3x-2}$$

Variable \lambda cannot be equal to any of the values \frac{4}{3},\frac{3}{2},2.

$$\lambda =\frac{4x-3}{3x-2}\text{, }\lambda \neq \frac{4}{3}\text{ and }\lambda \neq \frac{3}{2}\text{ and }\lambda \neq 2$$

Solve for x

$$x=-\frac{2\lambda -3}{4-3\lambda }$$

$$\lambda \neq \frac{4}{3}\text{ and }\lambda \neq \frac{3}{2}\text{ and }\lambda \neq 2$$