Solve 4-sqrt{2}=a+bsqrt{2}

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

Solve for b

$$b=-\frac{\sqrt{2}\left(a+\sqrt{2}-4\right)}{2}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

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

Subtract a from both sides.

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

The equation is in standard form.

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

Divide both sides by \sqrt{2}.

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

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

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

Divide 4-\sqrt{2}-a by \sqrt{2}.

$$b=-\frac{\sqrt{2}a}{2}+2\sqrt{2}-1$$

Solve for a

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