Solve x-1/6x^2+5x-6+2x-2/8x^2+2x-15=3x-3/12x^2-23x+10*lambda

$$\frac { x - 1 } { 6 x ^ { 2 } + 5 x - 6 } + \frac { 2 x - 2 } { 8 x ^ { 2 } + 2 x - 15 } = \frac { 3 x - 3 } { 12 x ^ { 2 } - 23 x + 10 } \cdot \lambda$$

Solve for λ (complex solution)

$$\left\{\begin{matrix}\lambda =-\frac{9-10x}{3\left(2x+3\right)}\text{, }&x\neq -\frac{3}{2}\text{ and }x\neq \frac{2}{3}\text{ and }x\neq \frac{5}{4}\\\lambda \in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6\left(\frac{1}{2}x-\frac{1}{3}\right)\left(4x-5\right)\left(2x+3\right), the least common multiple of 6x^{2}+5x-6,8x^{2}+2x-15,12x^{2}-23x+10.

$$\left(4x-5\right)\left(x-1\right)+\left(-2+3x\right)\left(2x-2\right)=\left(2x+3\right)\left(3x-3\right)\lambda$$

Use the distributive property to multiply 4x-5 by x-1 and combine like terms.

$$4x^{2}-9x+5+\left(-2+3x\right)\left(2x-2\right)=\left(2x+3\right)\left(3x-3\right)\lambda$$

Use the distributive property to multiply -2+3x by 2x-2 and combine like terms.

$$4x^{2}-9x+5-10x+4+6x^{2}=\left(2x+3\right)\left(3x-3\right)\lambda$$

Combine -9x and -10x to get -19x.

$$4x^{2}-19x+5+4+6x^{2}=\left(2x+3\right)\left(3x-3\right)\lambda$$

Add 5 and 4 to get 9.

$$4x^{2}-19x+9+6x^{2}=\left(2x+3\right)\left(3x-3\right)\lambda$$

Combine 4x^{2} and 6x^{2} to get 10x^{2}.

$$10x^{2}-19x+9=\left(2x+3\right)\left(3x-3\right)\lambda$$

Use the distributive property to multiply 2x+3 by 3x-3 and combine like terms.

$$10x^{2}-19x+9=\left(6x^{2}+3x-9\right)\lambda$$

Use the distributive property to multiply 6x^{2}+3x-9 by \lambda .

$$10x^{2}-19x+9=6x^{2}\lambda +3x\lambda -9\lambda$$

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

$$6x^{2}\lambda +3x\lambda -9\lambda =10x^{2}-19x+9$$

Combine all terms containing \lambda .

$$\left(6x^{2}+3x-9\right)\lambda =10x^{2}-19x+9$$

Divide both sides by 6x^{2}+3x-9.

$$\frac{\left(6x^{2}+3x-9\right)\lambda }{6x^{2}+3x-9}=\frac{\left(x-1\right)\left(10x-9\right)}{6x^{2}+3x-9}$$

Dividing by 6x^{2}+3x-9 undoes the multiplication by 6x^{2}+3x-9.

$$\lambda =\frac{\left(x-1\right)\left(10x-9\right)}{6x^{2}+3x-9}$$

Divide \left(-9+10x\right)\left(-1+x\right) by 6x^{2}+3x-9.

$$\lambda =\frac{10x-9}{3\left(2x+3\right)}$$

Solve for λ

$$\left\{\begin{matrix}\lambda =-\frac{9-10x}{3\left(2x+3\right)}\text{, }&x\neq \frac{2}{3}\text{ and }x\neq -\frac{3}{2}\text{ and }x\neq \frac{5}{4}\\\lambda \in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6\left(\frac{1}{2}x-\frac{1}{3}\right)\left(4x-5\right)\left(2x+3\right), the least common multiple of 6x^{2}+5x-6,8x^{2}+2x-15,12x^{2}-23x+10.

$$\left(4x-5\right)\left(x-1\right)+\left(-2+3x\right)\left(2x-2\right)=\left(2x+3\right)\left(3x-3\right)\lambda$$

Use the distributive property to multiply 4x-5 by x-1 and combine like terms.

$$4x^{2}-9x+5+\left(-2+3x\right)\left(2x-2\right)=\left(2x+3\right)\left(3x-3\right)\lambda$$

Use the distributive property to multiply -2+3x by 2x-2 and combine like terms.

$$4x^{2}-9x+5-10x+4+6x^{2}=\left(2x+3\right)\left(3x-3\right)\lambda$$

Combine -9x and -10x to get -19x.

$$4x^{2}-19x+5+4+6x^{2}=\left(2x+3\right)\left(3x-3\right)\lambda$$

Add 5 and 4 to get 9.

$$4x^{2}-19x+9+6x^{2}=\left(2x+3\right)\left(3x-3\right)\lambda$$

Combine 4x^{2} and 6x^{2} to get 10x^{2}.

$$10x^{2}-19x+9=\left(2x+3\right)\left(3x-3\right)\lambda$$

Use the distributive property to multiply 2x+3 by 3x-3 and combine like terms.

$$10x^{2}-19x+9=\left(6x^{2}+3x-9\right)\lambda$$

Use the distributive property to multiply 6x^{2}+3x-9 by \lambda .

$$10x^{2}-19x+9=6x^{2}\lambda +3x\lambda -9\lambda$$

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

$$6x^{2}\lambda +3x\lambda -9\lambda =10x^{2}-19x+9$$

Combine all terms containing \lambda .

$$\left(6x^{2}+3x-9\right)\lambda =10x^{2}-19x+9$$

Divide both sides by 6x^{2}+3x-9.

$$\frac{\left(6x^{2}+3x-9\right)\lambda }{6x^{2}+3x-9}=\frac{\left(x-1\right)\left(10x-9\right)}{6x^{2}+3x-9}$$

Dividing by 6x^{2}+3x-9 undoes the multiplication by 6x^{2}+3x-9.

$$\lambda =\frac{\left(x-1\right)\left(10x-9\right)}{6x^{2}+3x-9}$$

Divide \left(-9+10x\right)\left(-1+x\right) by 6x^{2}+3x-9.

$$\lambda =\frac{10x-9}{3\left(2x+3\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=\frac{9\left(\lambda +1\right)}{2\left(5-3\lambda \right)}\text{, }&\lambda \neq \frac{5}{3}\text{ and }\lambda \neq -\frac{7}{39}\text{ and }\lambda \neq \frac{7}{33}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=\frac{9\left(\lambda +1\right)}{2\left(5-3\lambda \right)}\text{, }&\lambda \neq -\frac{7}{39}\text{ and }\lambda \neq \frac{5}{3}\text{ and }\lambda \neq \frac{7}{33}\end{matrix}\right.$$