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

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

Solve for x

$$x=24-10\sqrt{6}\approx -0.494897428$$

Steps for Solving Linear Equation

Use the distributive property to multiply \sqrt{2} by x+3.

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

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

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

Subtract 3\sqrt{2} from both sides.

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

Combine -3\sqrt{2} and -3\sqrt{2} to get -6\sqrt{2}.

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

Add 2\sqrt{3} to both sides.

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

Combine 2\sqrt{3} and 2\sqrt{3} to get 4\sqrt{3}.

$$\sqrt{2}x+\sqrt{3}x=4\sqrt{3}-6\sqrt{2}$$

Combine all terms containing x.

$$\left(\sqrt{2}+\sqrt{3}\right)x=4\sqrt{3}-6\sqrt{2}$$

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

$$\frac{\left(\sqrt{2}+\sqrt{3}\right)x}{\sqrt{2}+\sqrt{3}}=\frac{4\sqrt{3}-6\sqrt{2}}{\sqrt{2}+\sqrt{3}}$$

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

$$x=\frac{4\sqrt{3}-6\sqrt{2}}{\sqrt{2}+\sqrt{3}}$$

Divide 4\sqrt{3}-6\sqrt{2} by \sqrt{2}+\sqrt{3}.

$$x=24-10\sqrt{6}$$