Solve (a+bsqrt{2})*(1+sqrt{2})=25+17sqrt{2}

$$( a + b \sqrt { 2 } ) \cdot ( 1 + \sqrt { 2 } ) = 25 + 17 \sqrt { 2 }$$

Solve for a

$$a=-\sqrt{2}\left(b-8\right)+9$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+b\sqrt{2} by 1+\sqrt{2}.

$$a+a\sqrt{2}+b\sqrt{2}+b\left(\sqrt{2}\right)^{2}=25+17\sqrt{2}$$

The square of \sqrt{2} is 2.

$$a+a\sqrt{2}+b\sqrt{2}+b\times 2=25+17\sqrt{2}$$

Subtract b\sqrt{2} from both sides.

$$a+a\sqrt{2}+b\times 2=25+17\sqrt{2}-b\sqrt{2}$$

Subtract b\times 2 from both sides.

$$a+a\sqrt{2}=25+17\sqrt{2}-b\sqrt{2}-b\times 2$$

Multiply -1 and 2 to get -2.

$$a+a\sqrt{2}=25+17\sqrt{2}-b\sqrt{2}-2b$$

Combine all terms containing a.

$$\left(1+\sqrt{2}\right)a=25+17\sqrt{2}-b\sqrt{2}-2b$$

The equation is in standard form.

$$\left(\sqrt{2}+1\right)a=-\sqrt{2}b-2b+17\sqrt{2}+25$$

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

$$\frac{\left(\sqrt{2}+1\right)a}{\sqrt{2}+1}=\frac{-\sqrt{2}b-2b+17\sqrt{2}+25}{\sqrt{2}+1}$$

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

$$a=\frac{-\sqrt{2}b-2b+17\sqrt{2}+25}{\sqrt{2}+1}$$

Divide 25+17\sqrt{2}-b\sqrt{2}-2b by 1+\sqrt{2}.

$$a=-\sqrt{2}b+8\sqrt{2}+9$$

Solve for b

$$b=-\frac{\sqrt{2}\left(a-8\sqrt{2}-9\right)}{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+b\sqrt{2} by 1+\sqrt{2}.

$$a+a\sqrt{2}+b\sqrt{2}+b\left(\sqrt{2}\right)^{2}=25+17\sqrt{2}$$

The square of \sqrt{2} is 2.

$$a+a\sqrt{2}+b\sqrt{2}+b\times 2=25+17\sqrt{2}$$

Subtract a from both sides.

$$a\sqrt{2}+b\sqrt{2}+b\times 2=25+17\sqrt{2}-a$$

Subtract a\sqrt{2} from both sides.

$$b\sqrt{2}+b\times 2=25+17\sqrt{2}-a-a\sqrt{2}$$

Reorder the terms.

$$\sqrt{2}b+2b=-\sqrt{2}a-a+17\sqrt{2}+25$$

Combine all terms containing b.

$$\left(\sqrt{2}+2\right)b=-\sqrt{2}a-a+17\sqrt{2}+25$$

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

$$\frac{\left(\sqrt{2}+2\right)b}{\sqrt{2}+2}=\frac{-\sqrt{2}a-a+17\sqrt{2}+25}{\sqrt{2}+2}$$

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

$$b=\frac{-\sqrt{2}a-a+17\sqrt{2}+25}{\sqrt{2}+2}$$

Divide -\sqrt{2}a-a+17\sqrt{2}+25 by \sqrt{2}+2.

$$b=\frac{\left(2-\sqrt{2}\right)\left(-\sqrt{2}a-a+17\sqrt{2}+25\right)}{2}$$