Solve 24-5x/x-2+8x-49/4-x=28/x-2-13

$$\frac { 24 - 5 x } { x - 2 } + \frac { 8 x - 49 } { 4 - x } = \frac { 28 } { x - 2 } - 13$$

Solve for x

$$x = -\frac{22}{3} = -7\frac{1}{3} \approx -7.333333333$$

Steps for Solving Linear Equation

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

$$\left(x-4\right)\left(24-5x\right)+\left(2-x\right)\left(8x-49\right)=\left(x-4\right)\times 28+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Use the distributive property to multiply x-4 by 24-5x and combine like terms.

$$44x-5x^{2}-96+\left(2-x\right)\left(8x-49\right)=\left(x-4\right)\times 28+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Use the distributive property to multiply 2-x by 8x-49 and combine like terms.

$$44x-5x^{2}-96+65x-98-8x^{2}=\left(x-4\right)\times 28+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Combine 44x and 65x to get 109x.

$$109x-5x^{2}-96-98-8x^{2}=\left(x-4\right)\times 28+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Subtract 98 from -96 to get -194.

$$109x-5x^{2}-194-8x^{2}=\left(x-4\right)\times 28+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Combine -5x^{2} and -8x^{2} to get -13x^{2}.

$$109x-13x^{2}-194=\left(x-4\right)\times 28+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Use the distributive property to multiply x-4 by 28.

$$109x-13x^{2}-194=28x-112+\left(x-4\right)\left(x-2\right)\left(-13\right)$$

Use the distributive property to multiply x-4 by x-2 and combine like terms.

$$109x-13x^{2}-194=28x-112+\left(x^{2}-6x+8\right)\left(-13\right)$$

Use the distributive property to multiply x^{2}-6x+8 by -13.

$$109x-13x^{2}-194=28x-112-13x^{2}+78x-104$$

Combine 28x and 78x to get 106x.

$$109x-13x^{2}-194=106x-112-13x^{2}-104$$

Subtract 104 from -112 to get -216.

$$109x-13x^{2}-194=106x-216-13x^{2}$$

Subtract 106x from both sides.

$$109x-13x^{2}-194-106x=-216-13x^{2}$$

Combine 109x and -106x to get 3x.

$$3x-13x^{2}-194=-216-13x^{2}$$

Add 13x^{2} to both sides.

$$3x-13x^{2}-194+13x^{2}=-216$$

Combine -13x^{2} and 13x^{2} to get 0.

$$3x-194=-216$$

Add 194 to both sides.

$$3x=-216+194$$

Add -216 and 194 to get -22.

$$3x=-22$$

Divide both sides by 3.

$$x=\frac{-22}{3}$$

Fraction \frac{-22}{3} can be rewritten as -\frac{22}{3} by extracting the negative sign.

$$x=-\frac{22}{3}$$