Solve -frac{11sqrt{15}+68}{53}*(frac{33sqrt{15}}{136}+3/2)+x=159/136

$$- \frac{ 11 \sqrt{ 15 } +68 }{ 53 } \times ( \frac{ 33 \sqrt{ 15 } }{ 136 } + \frac{ 3 }{ 2 } )+x= \frac{ 159 }{ 136 }$$

Solve for x

$$x = \frac{33 \sqrt{15} + 204}{53} \approx 6.2605368$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 7208, the least common multiple of 53,136,2.

$$7208\left(-\frac{11\sqrt{15}+68}{53}\right)\left(\frac{33\sqrt{15}}{136}+\frac{3}{2}\right)+7208x=8427$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 136 and 2 is 136. Multiply \frac{3}{2} times \frac{68}{68}.

$$7208\left(-\frac{11\sqrt{15}+68}{53}\right)\left(\frac{33\sqrt{15}}{136}+\frac{3\times 68}{136}\right)+7208x=8427$$

Since \frac{33\sqrt{15}}{136} and \frac{3\times 68}{136} have the same denominator, add them by adding their numerators.

$$7208\left(-\frac{11\sqrt{15}+68}{53}\right)\times \frac{33\sqrt{15}+3\times 68}{136}+7208x=8427$$

Do the multiplications in 33\sqrt{15}+3\times 68.

$$7208\left(-\frac{11\sqrt{15}+68}{53}\right)\times \frac{33\sqrt{15}+204}{136}+7208x=8427$$

Cancel out 53, the greatest common factor in 7208 and 53.

$$-136\left(11\sqrt{15}+68\right)\times \frac{33\sqrt{15}+204}{136}+7208x=8427$$

Express -136\left(11\sqrt{15}+68\right)\times \frac{33\sqrt{15}+204}{136} as a single fraction.

$$\frac{-136\left(11\sqrt{15}+68\right)\left(33\sqrt{15}+204\right)}{136}+7208x=8427$$

Cancel out 136 and 136.

$$-\left(11\sqrt{15}+68\right)\left(33\sqrt{15}+204\right)+7208x=8427$$

Use the distributive property to multiply -1 by 11\sqrt{15}+68.

$$\left(-11\sqrt{15}-68\right)\left(33\sqrt{15}+204\right)+7208x=8427$$

Apply the distributive property by multiplying each term of -11\sqrt{15}-68 by each term of 33\sqrt{15}+204.

$$-363\left(\sqrt{15}\right)^{2}-2244\sqrt{15}-2244\sqrt{15}-13872+7208x=8427$$

The square of \sqrt{15} is 15.

$$-363\times 15-2244\sqrt{15}-2244\sqrt{15}-13872+7208x=8427$$

Multiply -363 and 15 to get -5445.

$$-5445-2244\sqrt{15}-2244\sqrt{15}-13872+7208x=8427$$

Combine -2244\sqrt{15} and -2244\sqrt{15} to get -4488\sqrt{15}.

$$-5445-4488\sqrt{15}-13872+7208x=8427$$

Subtract 13872 from -5445 to get -19317.

$$-19317-4488\sqrt{15}+7208x=8427$$

Add 19317 to both sides.

$$-4488\sqrt{15}+7208x=8427+19317$$

Add 8427 and 19317 to get 27744.

$$-4488\sqrt{15}+7208x=27744$$

Add 4488\sqrt{15} to both sides.

$$7208x=27744+4488\sqrt{15}$$

The equation is in standard form.

$$7208x=4488\sqrt{15}+27744$$

Divide both sides by 7208.

$$\frac{7208x}{7208}=\frac{4488\sqrt{15}+27744}{7208}$$

Dividing by 7208 undoes the multiplication by 7208.

$$x=\frac{4488\sqrt{15}+27744}{7208}$$

Divide 27744+4488\sqrt{15} by 7208.

$$x=\frac{33\sqrt{15}+204}{53}$$