Solve (1-sqrt{2})(-(sqrt{2}+1)sqrt{3}x+x)=100sqrt{2}

$$(1- \sqrt{ 2 } )(-( \sqrt{ 2 } +1) \sqrt{ 3 } x+x)=100 \sqrt{ 2 }$$

Solve for x

$$x = \frac{100 \sqrt{2} {(\sqrt{2} + 1)}}{1 - \sqrt{3} - \sqrt{6}} \approx 107.313218497$$

Steps for Solving Linear Equation

To find the opposite of \sqrt{2}+1, find the opposite of each term.

$$\left(1-\sqrt{2}\right)\left(\left(-\sqrt{2}-1\right)\sqrt{3}x+x\right)=100\sqrt{2}$$

Use the distributive property to multiply -\sqrt{2}-1 by \sqrt{3}.

$$\left(1-\sqrt{2}\right)\left(\left(-\sqrt{2}\sqrt{3}-\sqrt{3}\right)x+x\right)=100\sqrt{2}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\left(1-\sqrt{2}\right)\left(\left(-\sqrt{6}-\sqrt{3}\right)x+x\right)=100\sqrt{2}$$

Use the distributive property to multiply -\sqrt{6}-\sqrt{3} by x.

$$\left(1-\sqrt{2}\right)\left(-\sqrt{6}x-\sqrt{3}x+x\right)=100\sqrt{2}$$

Apply the distributive property by multiplying each term of 1-\sqrt{2} by each term of -\sqrt{6}x-\sqrt{3}x+x.

$$-\sqrt{6}x-\sqrt{3}x+x+x\sqrt{2}\sqrt{6}+\sqrt{3}x\sqrt{2}-\sqrt{2}x=100\sqrt{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$-\sqrt{6}x-\sqrt{3}x+x+x\sqrt{2}\sqrt{2}\sqrt{3}+\sqrt{3}x\sqrt{2}-\sqrt{2}x=100\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-\sqrt{6}x-\sqrt{3}x+x+x\times 2\sqrt{3}+\sqrt{3}x\sqrt{2}-\sqrt{2}x=100\sqrt{2}$$

Combine -\sqrt{3}x and x\times 2\sqrt{3} to get \sqrt{3}x.

$$-\sqrt{6}x+\sqrt{3}x+x+\sqrt{3}x\sqrt{2}-\sqrt{2}x=100\sqrt{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$-\sqrt{6}x+\sqrt{3}x+x+\sqrt{6}x-\sqrt{2}x=100\sqrt{2}$$

Combine -\sqrt{6}x and \sqrt{6}x to get 0.

$$\sqrt{3}x+x-\sqrt{2}x=100\sqrt{2}$$

Combine all terms containing x.

$$\left(\sqrt{3}+1-\sqrt{2}\right)x=100\sqrt{2}$$

Divide both sides by \sqrt{3}+1-\sqrt{2}.

$$\frac{\left(\sqrt{3}+1-\sqrt{2}\right)x}{\sqrt{3}+1-\sqrt{2}}=\frac{100\sqrt{2}}{\sqrt{3}+1-\sqrt{2}}$$

Dividing by \sqrt{3}+1-\sqrt{2} undoes the multiplication by \sqrt{3}+1-\sqrt{2}.

$$x=\frac{100\sqrt{2}}{\sqrt{3}+1-\sqrt{2}}$$

Divide 100\sqrt{2} by \sqrt{3}+1-\sqrt{2}.

$$x=50\sqrt{2}+50\sqrt{3}-50$$