Solve 3/lambda-6+7/x-2=10/lambda-4

$$\frac { 3 } { \lambda - 6 } + \frac { 7 } { x - 2 } = \frac { 10 } { \lambda - 4 }$$

Solve for x

$$x=-\frac{7\lambda ^{2}-56\lambda +72}{48-7\lambda }$$

$$\lambda \neq 4\text{ and }\lambda \neq 6\text{ and }\lambda \neq \frac{48}{7}$$

Steps for Solving Linear Equation

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(\lambda -6\right)\left(\lambda -4\right), the least common multiple of \lambda -6,x-2,\lambda -4.

$$\left(x-2\right)\left(\lambda -4\right)\times 3+\left(\lambda -6\right)\left(\lambda -4\right)\times 7=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Use the distributive property to multiply x-2 by \lambda -4.

$$\left(x\lambda -4x-2\lambda +8\right)\times 3+\left(\lambda -6\right)\left(\lambda -4\right)\times 7=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Use the distributive property to multiply x\lambda -4x-2\lambda +8 by 3.

$$3x\lambda -12x-6\lambda +24+\left(\lambda -6\right)\left(\lambda -4\right)\times 7=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Use the distributive property to multiply \lambda -6 by \lambda -4 and combine like terms.

$$3x\lambda -12x-6\lambda +24+\left(\lambda ^{2}-10\lambda +24\right)\times 7=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Use the distributive property to multiply \lambda ^{2}-10\lambda +24 by 7.

$$3x\lambda -12x-6\lambda +24+7\lambda ^{2}-70\lambda +168=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Combine -6\lambda and -70\lambda to get -76\lambda .

$$3x\lambda -12x-76\lambda +24+7\lambda ^{2}+168=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Add 24 and 168 to get 192.

$$3x\lambda -12x-76\lambda +192+7\lambda ^{2}=\left(x-2\right)\left(\lambda -6\right)\times 10$$

Use the distributive property to multiply x-2 by \lambda -6.

$$3x\lambda -12x-76\lambda +192+7\lambda ^{2}=\left(x\lambda -6x-2\lambda +12\right)\times 10$$

Use the distributive property to multiply x\lambda -6x-2\lambda +12 by 10.

$$3x\lambda -12x-76\lambda +192+7\lambda ^{2}=10x\lambda -60x-20\lambda +120$$

Subtract 10x\lambda from both sides.

$$3x\lambda -12x-76\lambda +192+7\lambda ^{2}-10x\lambda =-60x-20\lambda +120$$

Combine 3x\lambda and -10x\lambda to get -7x\lambda .

$$-7x\lambda -12x-76\lambda +192+7\lambda ^{2}=-60x-20\lambda +120$$

Add 60x to both sides.

$$-7x\lambda -12x-76\lambda +192+7\lambda ^{2}+60x=-20\lambda +120$$

Combine -12x and 60x to get 48x.

$$-7x\lambda +48x-76\lambda +192+7\lambda ^{2}=-20\lambda +120$$

Add 76\lambda to both sides.

$$-7x\lambda +48x+192+7\lambda ^{2}=-20\lambda +120+76\lambda$$

Combine -20\lambda and 76\lambda to get 56\lambda .

$$-7x\lambda +48x+192+7\lambda ^{2}=56\lambda +120$$

Subtract 192 from both sides.

$$-7x\lambda +48x+7\lambda ^{2}=56\lambda +120-192$$

Subtract 192 from 120 to get -72.

$$-7x\lambda +48x+7\lambda ^{2}=56\lambda -72$$

Subtract 7\lambda ^{2} from both sides.

$$-7x\lambda +48x=56\lambda -72-7\lambda ^{2}$$

Combine all terms containing x.

$$\left(-7\lambda +48\right)x=56\lambda -72-7\lambda ^{2}$$

The equation is in standard form.

$$\left(48-7\lambda \right)x=-7\lambda ^{2}+56\lambda -72$$

Divide both sides by -7\lambda +48.

$$\frac{\left(48-7\lambda \right)x}{48-7\lambda }=\frac{-7\lambda ^{2}+56\lambda -72}{48-7\lambda }$$

Dividing by -7\lambda +48 undoes the multiplication by -7\lambda +48.

$$x=\frac{-7\lambda ^{2}+56\lambda -72}{48-7\lambda }$$

Variable x cannot be equal to 2.

$$x=\frac{-7\lambda ^{2}+56\lambda -72}{48-7\lambda }\text{, }x\neq 2$$

Solve for λ (complex solution)

$$\lambda =\frac{\sqrt{49x^{2}-560x+1120}}{14}+\frac{x}{2}+4$$

$$\lambda =-\frac{\sqrt{49x^{2}-560x+1120}}{14}+\frac{x}{2}+4\text{, }x\neq 2$$

Solve for λ

$$\lambda =\frac{\sqrt{49x^{2}-560x+1120}}{14}+\frac{x}{2}+4$$

$$\lambda =-\frac{\sqrt{49x^{2}-560x+1120}}{14}+\frac{x}{2}+4\text{, }\left(x\neq 2\text{ and }x\leq \frac{40-4\sqrt{30}}{7}\right)\text{ or }x\geq \frac{4\sqrt{30}+40}{7}$$