Solve 3/x+3-2/2x-5=6/3x-13

$$\frac { 3 } { x + 3 } - \frac { 2 } { 2 x - 5 } = \frac { 6 } { 3 x - 13 }$$

Solve for x

$$x=3$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,\frac{5}{2},\frac{13}{3} since division by zero is not defined. Multiply both sides of the equation by 3\left(2x-5\right)\left(3x-13\right)\left(\frac{1}{3}x+1\right), the least common multiple of x+3,2x-5,3x-13.

$$\left(2x-5\right)\left(3x-13\right)\times 3-\left(3x-13\right)\left(3+x\right)\times 2=\left(2x-5\right)\left(3+x\right)\times 6$$

Use the distributive property to multiply 2x-5 by 3x-13 and combine like terms.

$$\left(6x^{2}-41x+65\right)\times 3-\left(3x-13\right)\left(3+x\right)\times 2=\left(2x-5\right)\left(3+x\right)\times 6$$

Use the distributive property to multiply 6x^{2}-41x+65 by 3.

$$18x^{2}-123x+195-\left(3x-13\right)\left(3+x\right)\times 2=\left(2x-5\right)\left(3+x\right)\times 6$$

Use the distributive property to multiply 3x-13 by 3+x and combine like terms.

$$18x^{2}-123x+195-\left(-4x+3x^{2}-39\right)\times 2=\left(2x-5\right)\left(3+x\right)\times 6$$

Use the distributive property to multiply -4x+3x^{2}-39 by 2.

$$18x^{2}-123x+195-\left(-8x+6x^{2}-78\right)=\left(2x-5\right)\left(3+x\right)\times 6$$

To find the opposite of -8x+6x^{2}-78, find the opposite of each term.

$$18x^{2}-123x+195+8x-6x^{2}+78=\left(2x-5\right)\left(3+x\right)\times 6$$

Combine -123x and 8x to get -115x.

$$18x^{2}-115x+195-6x^{2}+78=\left(2x-5\right)\left(3+x\right)\times 6$$

Combine 18x^{2} and -6x^{2} to get 12x^{2}.

$$12x^{2}-115x+195+78=\left(2x-5\right)\left(3+x\right)\times 6$$

Add 195 and 78 to get 273.

$$12x^{2}-115x+273=\left(2x-5\right)\left(3+x\right)\times 6$$

Use the distributive property to multiply 2x-5 by 3+x and combine like terms.

$$12x^{2}-115x+273=\left(x+2x^{2}-15\right)\times 6$$

Use the distributive property to multiply x+2x^{2}-15 by 6.

$$12x^{2}-115x+273=6x+12x^{2}-90$$

Subtract 6x from both sides.

$$12x^{2}-115x+273-6x=12x^{2}-90$$

Combine -115x and -6x to get -121x.

$$12x^{2}-121x+273=12x^{2}-90$$

Subtract 12x^{2} from both sides.

$$12x^{2}-121x+273-12x^{2}=-90$$

Combine 12x^{2} and -12x^{2} to get 0.

$$-121x+273=-90$$

Subtract 273 from both sides.

$$-121x=-90-273$$

Subtract 273 from -90 to get -363.

$$-121x=-363$$

Divide both sides by -121.

$$x=\frac{-363}{-121}$$

Divide -363 by -121 to get 3.

$$x=3$$