Solve (a+3sqrt{2})(5-4sqrt{2})=bsqrt{2}-19,f

$$( a + 3 \sqrt { 2 } ) ( 5 - 4 \sqrt { 2 } ) = b \sqrt { 2 } - 19 , f$$

Solve for a

$$a=\frac{-5\sqrt{2}\left(b-11\right)-8b+95}{7}$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+3\sqrt{2} by 5-4\sqrt{2}.

$$5a-4a\sqrt{2}+15\sqrt{2}-12\left(\sqrt{2}\right)^{2}=b\sqrt{2}-19$$

The square of \sqrt{2} is 2.

$$5a-4a\sqrt{2}+15\sqrt{2}-12\times 2=b\sqrt{2}-19$$

Multiply -12 and 2 to get -24.

$$5a-4a\sqrt{2}+15\sqrt{2}-24=b\sqrt{2}-19$$

Subtract 15\sqrt{2} from both sides.

$$5a-4a\sqrt{2}-24=b\sqrt{2}-19-15\sqrt{2}$$

Add 24 to both sides.

$$5a-4a\sqrt{2}=b\sqrt{2}-19-15\sqrt{2}+24$$

Add -19 and 24 to get 5.

$$5a-4a\sqrt{2}=b\sqrt{2}+5-15\sqrt{2}$$

Combine all terms containing a.

$$\left(5-4\sqrt{2}\right)a=b\sqrt{2}+5-15\sqrt{2}$$

The equation is in standard form.

$$\left(5-4\sqrt{2}\right)a=\sqrt{2}b+5-15\sqrt{2}$$

Divide both sides by 5-4\sqrt{2}.

$$\frac{\left(5-4\sqrt{2}\right)a}{5-4\sqrt{2}}=\frac{\sqrt{2}b+5-15\sqrt{2}}{5-4\sqrt{2}}$$

Dividing by 5-4\sqrt{2} undoes the multiplication by 5-4\sqrt{2}.

$$a=\frac{\sqrt{2}b+5-15\sqrt{2}}{5-4\sqrt{2}}$$

Divide b\sqrt{2}+5-15\sqrt{2} by 5-4\sqrt{2}.

$$a=\frac{-5\sqrt{2}b-8b+55\sqrt{2}+95}{7}$$

Solve for b

$$b=\frac{5\sqrt{2}a}{2}-4a-\frac{5\sqrt{2}}{2}+15$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+3\sqrt{2} by 5-4\sqrt{2}.

$$5a-4a\sqrt{2}+15\sqrt{2}-12\left(\sqrt{2}\right)^{2}=b\sqrt{2}-19$$

The square of \sqrt{2} is 2.

$$5a-4a\sqrt{2}+15\sqrt{2}-12\times 2=b\sqrt{2}-19$$

Multiply -12 and 2 to get -24.

$$5a-4a\sqrt{2}+15\sqrt{2}-24=b\sqrt{2}-19$$

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

$$b\sqrt{2}-19=5a-4a\sqrt{2}+15\sqrt{2}-24$$

Add 19 to both sides.

$$b\sqrt{2}=5a-4a\sqrt{2}+15\sqrt{2}-24+19$$

Add -24 and 19 to get -5.

$$b\sqrt{2}=5a-4a\sqrt{2}+15\sqrt{2}-5$$

The equation is in standard form.

$$\sqrt{2}b=-4\sqrt{2}a+5a+15\sqrt{2}-5$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}b}{\sqrt{2}}=\frac{-4\sqrt{2}a+5a+15\sqrt{2}-5}{\sqrt{2}}$$

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

$$b=\frac{-4\sqrt{2}a+5a+15\sqrt{2}-5}{\sqrt{2}}$$

Divide 5a-4a\sqrt{2}+15\sqrt{2}-5 by \sqrt{2}.

$$b=\frac{5\sqrt{2}a}{2}-4a-\frac{5\sqrt{2}}{2}+15$$