Solve a+bsqrt{2}=3(1-sqrt{2})+4sqrt{2}

$$a + b \sqrt { 2 } = 3 ( 1 - \sqrt { 2 } ) + 4 \sqrt { 2 }$$

Solve for b

$$b=-\frac{\sqrt{2}\left(a-\sqrt{2}-3\right)}{2}$$

Steps for Solving Linear Equation

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

$$a+b\sqrt{2}=3-3\sqrt{2}+4\sqrt{2}$$

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

$$a+b\sqrt{2}=3+\sqrt{2}$$

Subtract a from both sides.

$$b\sqrt{2}=3+\sqrt{2}-a$$

The equation is in standard form.

$$\sqrt{2}b=-a+\sqrt{2}+3$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}b}{\sqrt{2}}=\frac{-a+\sqrt{2}+3}{\sqrt{2}}$$

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

$$b=\frac{-a+\sqrt{2}+3}{\sqrt{2}}$$

Divide 3+\sqrt{2}-a by \sqrt{2}.

$$b=\frac{\sqrt{2}\left(-a+\sqrt{2}+3\right)}{2}$$

Solve for a

$$a=-\sqrt{2}\left(b-1\right)+3$$