Solve frac{10-sqrt{18}}{sqrt{2}}=a+bsqrt{2}

$$\frac { 10 - \sqrt { 18 } } { \sqrt { 2 } } = a + b \sqrt { 2 }$$

Solve for b

$$b=-\frac{\sqrt{2}\left(a+3-5\sqrt{2}\right)}{2}$$

Steps for Solving Linear Equation

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{10-3\sqrt{2}}{\sqrt{2}}=a+b\sqrt{2}$$

Rationalize the denominator of \frac{10-3\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(10-3\sqrt{2}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}=a+b\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{\left(10-3\sqrt{2}\right)\sqrt{2}}{2}=a+b\sqrt{2}$$

Use the distributive property to multiply 10-3\sqrt{2} by \sqrt{2}.

$$\frac{10\sqrt{2}-3\left(\sqrt{2}\right)^{2}}{2}=a+b\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{10\sqrt{2}-3\times 2}{2}=a+b\sqrt{2}$$

Multiply -3 and 2 to get -6.

$$\frac{10\sqrt{2}-6}{2}=a+b\sqrt{2}$$

Divide each term of 10\sqrt{2}-6 by 2 to get 5\sqrt{2}-3.

$$5\sqrt{2}-3=a+b\sqrt{2}$$

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

$$a+b\sqrt{2}=5\sqrt{2}-3$$

Subtract a from both sides.

$$b\sqrt{2}=5\sqrt{2}-3-a$$

The equation is in standard form.

$$\sqrt{2}b=-a+5\sqrt{2}-3$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}b}{\sqrt{2}}=\frac{-a+5\sqrt{2}-3}{\sqrt{2}}$$

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

$$b=\frac{-a+5\sqrt{2}-3}{\sqrt{2}}$$

Divide 5\sqrt{2}-a-3 by \sqrt{2}.

$$b=\frac{\sqrt{2}\left(-a+5\sqrt{2}-3\right)}{2}$$

Solve for a

$$a=-\sqrt{2}b+5\sqrt{2}-3$$