Solve 5/x^2-7x+12=2/x-3+5/x-4quadtext{LCD}=

$$\frac { 5 } { x ^ { 2 } - 7 x + 12 } = \frac { 2 } { x - 3 } + \frac { 5 } { x - 4 } \quad \text { LCD } =$$

Solve for C (complex solution)

$$\left\{\begin{matrix}C=-\frac{13-2x}{5DL\left(3-x\right)}\text{, }&x\neq 3\text{ and }D\neq 0\text{ and }L\neq 0\text{ and }x\neq 4\\C\in \mathrm{C}\text{, }&\left(L=0\text{ or }D=0\right)\text{ and }x=\frac{13}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x^{2}-7x+12,x-3,x-4.

$$5=\left(x-4\right)\times 2+\left(x-3\right)\times 5LCD$$

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

$$5=2x-8+\left(x-3\right)\times 5LCD$$

Use the distributive property to multiply x-3 by 5.

$$5=2x-8+\left(5x-15\right)LCD$$

Use the distributive property to multiply 5x-15 by L.

$$5=2x-8+\left(5xL-15L\right)CD$$

Use the distributive property to multiply 5xL-15L by C.

$$5=2x-8+\left(5xLC-15LC\right)D$$

Use the distributive property to multiply 5xLC-15LC by D.

$$5=2x-8+5xLCD-15LCD$$

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

$$2x-8+5xLCD-15LCD=5$$

Subtract 2x from both sides.

$$-8+5xLCD-15LCD=5-2x$$

Add 8 to both sides.

$$5xLCD-15LCD=5-2x+8$$

Add 5 and 8 to get 13.

$$5xLCD-15LCD=13-2x$$

Combine all terms containing C.

$$\left(5xLD-15LD\right)C=13-2x$$

The equation is in standard form.

$$\left(5DLx-15DL\right)C=13-2x$$

Divide both sides by 5xLD-15LD.

$$\frac{\left(5DLx-15DL\right)C}{5DLx-15DL}=\frac{13-2x}{5DLx-15DL}$$

Dividing by 5xLD-15LD undoes the multiplication by 5xLD-15LD.

$$C=\frac{13-2x}{5DLx-15DL}$$

Divide 13-2x by 5xLD-15LD.

$$C=\frac{13-2x}{5DL\left(x-3\right)}$$

Solve for D (complex solution)

$$\left\{\begin{matrix}D=-\frac{13-2x}{5CL\left(3-x\right)}\text{, }&x\neq 3\text{ and }C\neq 0\text{ and }L\neq 0\text{ and }x\neq 4\\D\in \mathrm{C}\text{, }&\left(L=0\text{ or }C=0\right)\text{ and }x=\frac{13}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x^{2}-7x+12,x-3,x-4.

$$5=\left(x-4\right)\times 2+\left(x-3\right)\times 5LCD$$

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

$$5=2x-8+\left(x-3\right)\times 5LCD$$

Use the distributive property to multiply x-3 by 5.

$$5=2x-8+\left(5x-15\right)LCD$$

Use the distributive property to multiply 5x-15 by L.

$$5=2x-8+\left(5xL-15L\right)CD$$

Use the distributive property to multiply 5xL-15L by C.

$$5=2x-8+\left(5xLC-15LC\right)D$$

Use the distributive property to multiply 5xLC-15LC by D.

$$5=2x-8+5xLCD-15LCD$$

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

$$2x-8+5xLCD-15LCD=5$$

Subtract 2x from both sides.

$$-8+5xLCD-15LCD=5-2x$$

Add 8 to both sides.

$$5xLCD-15LCD=5-2x+8$$

Add 5 and 8 to get 13.

$$5xLCD-15LCD=13-2x$$

Combine all terms containing D.

$$\left(5xLC-15LC\right)D=13-2x$$

The equation is in standard form.

$$\left(5CLx-15CL\right)D=13-2x$$

Divide both sides by 5xLC-15LC.

$$\frac{\left(5CLx-15CL\right)D}{5CLx-15CL}=\frac{13-2x}{5CLx-15CL}$$

Dividing by 5xLC-15LC undoes the multiplication by 5xLC-15LC.

$$D=\frac{13-2x}{5CLx-15CL}$$

Divide 13-2x by 5xLC-15LC.

$$D=\frac{13-2x}{5CL\left(x-3\right)}$$

Solve for C

$$\left\{\begin{matrix}C=-\frac{13-2x}{5DL\left(3-x\right)}\text{, }&x\neq 3\text{ and }D\neq 0\text{ and }L\neq 0\text{ and }x\neq 4\\C\in \mathrm{R}\text{, }&\left(L=0\text{ or }D=0\right)\text{ and }x=\frac{13}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x^{2}-7x+12,x-3,x-4.

$$5=\left(x-4\right)\times 2+\left(x-3\right)\times 5LCD$$

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

$$5=2x-8+\left(x-3\right)\times 5LCD$$

Use the distributive property to multiply x-3 by 5.

$$5=2x-8+\left(5x-15\right)LCD$$

Use the distributive property to multiply 5x-15 by L.

$$5=2x-8+\left(5xL-15L\right)CD$$

Use the distributive property to multiply 5xL-15L by C.

$$5=2x-8+\left(5xLC-15LC\right)D$$

Use the distributive property to multiply 5xLC-15LC by D.

$$5=2x-8+5xLCD-15LCD$$

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

$$2x-8+5xLCD-15LCD=5$$

Subtract 2x from both sides.

$$-8+5xLCD-15LCD=5-2x$$

Add 8 to both sides.

$$5xLCD-15LCD=5-2x+8$$

Add 5 and 8 to get 13.

$$5xLCD-15LCD=13-2x$$

Combine all terms containing C.

$$\left(5xLD-15LD\right)C=13-2x$$

The equation is in standard form.

$$\left(5DLx-15DL\right)C=13-2x$$

Divide both sides by 5xLD-15LD.

$$\frac{\left(5DLx-15DL\right)C}{5DLx-15DL}=\frac{13-2x}{5DLx-15DL}$$

Dividing by 5xLD-15LD undoes the multiplication by 5xLD-15LD.

$$C=\frac{13-2x}{5DLx-15DL}$$

Divide -2x+13 by 5xLD-15LD.

$$C=\frac{13-2x}{5DL\left(x-3\right)}$$

Solve for D

$$\left\{\begin{matrix}D=-\frac{13-2x}{5CL\left(3-x\right)}\text{, }&x\neq 3\text{ and }C\neq 0\text{ and }L\neq 0\text{ and }x\neq 4\\D\in \mathrm{R}\text{, }&\left(L=0\text{ or }C=0\right)\text{ and }x=\frac{13}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x^{2}-7x+12,x-3,x-4.

$$5=\left(x-4\right)\times 2+\left(x-3\right)\times 5LCD$$

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

$$5=2x-8+\left(x-3\right)\times 5LCD$$

Use the distributive property to multiply x-3 by 5.

$$5=2x-8+\left(5x-15\right)LCD$$

Use the distributive property to multiply 5x-15 by L.

$$5=2x-8+\left(5xL-15L\right)CD$$

Use the distributive property to multiply 5xL-15L by C.

$$5=2x-8+\left(5xLC-15LC\right)D$$

Use the distributive property to multiply 5xLC-15LC by D.

$$5=2x-8+5xLCD-15LCD$$

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

$$2x-8+5xLCD-15LCD=5$$

Subtract 2x from both sides.

$$-8+5xLCD-15LCD=5-2x$$

Add 8 to both sides.

$$5xLCD-15LCD=5-2x+8$$

Add 5 and 8 to get 13.

$$5xLCD-15LCD=13-2x$$

Combine all terms containing D.

$$\left(5xLC-15LC\right)D=13-2x$$

The equation is in standard form.

$$\left(5CLx-15CL\right)D=13-2x$$

Divide both sides by 5xLC-15LC.

$$\frac{\left(5CLx-15CL\right)D}{5CLx-15CL}=\frac{13-2x}{5CLx-15CL}$$

Dividing by 5xLC-15LC undoes the multiplication by 5xLC-15LC.

$$D=\frac{13-2x}{5CLx-15CL}$$

Divide -2x+13 by 5xLC-15LC.

$$D=\frac{13-2x}{5CL\left(x-3\right)}$$