Solve a-8/sqrt{2+3}=a-15

$$\frac { a - 8 } { \sqrt { 2 } + 3 } = a - 15$$

Solve for a

$$a=-\frac{7\sqrt{2}}{2}+22\approx 17.050252532$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{a-8}{\sqrt{2}+3} by multiplying numerator and denominator by \sqrt{2}-3.

$$\frac{\left(a-8\right)\left(\sqrt{2}-3\right)}{\left(\sqrt{2}+3\right)\left(\sqrt{2}-3\right)}=a-15$$

Consider \left(\sqrt{2}+3\right)\left(\sqrt{2}-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(a-8\right)\left(\sqrt{2}-3\right)}{\left(\sqrt{2}\right)^{2}-3^{2}}=a-15$$

Square \sqrt{2}. Square 3.

$$\frac{\left(a-8\right)\left(\sqrt{2}-3\right)}{2-9}=a-15$$

Subtract 9 from 2 to get -7.

$$\frac{\left(a-8\right)\left(\sqrt{2}-3\right)}{-7}=a-15$$

Apply the distributive property by multiplying each term of a-8 by each term of \sqrt{2}-3.

$$\frac{a\sqrt{2}-3a-8\sqrt{2}+24}{-7}=a-15$$

Subtract a from both sides.

$$\frac{a\sqrt{2}-3a-8\sqrt{2}+24}{-7}-a=-15$$

Multiply both sides of the equation by -7.

$$a\sqrt{2}-3a-8\sqrt{2}+24+7a=105$$

Reorder the terms.

$$\sqrt{2}a-3a+7a-8\sqrt{2}+24=105$$

Combine -3a and 7a to get 4a.

$$\sqrt{2}a+4a-8\sqrt{2}+24=105$$

Add 8\sqrt{2} to both sides.

$$\sqrt{2}a+4a+24=105+8\sqrt{2}$$

Subtract 24 from both sides.

$$\sqrt{2}a+4a=105+8\sqrt{2}-24$$

Subtract 24 from 105 to get 81.

$$\sqrt{2}a+4a=81+8\sqrt{2}$$

Combine all terms containing a.

$$\left(\sqrt{2}+4\right)a=81+8\sqrt{2}$$

The equation is in standard form.

$$\left(\sqrt{2}+4\right)a=8\sqrt{2}+81$$

Divide both sides by \sqrt{2}+4.

$$\frac{\left(\sqrt{2}+4\right)a}{\sqrt{2}+4}=\frac{8\sqrt{2}+81}{\sqrt{2}+4}$$

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

$$a=\frac{8\sqrt{2}+81}{\sqrt{2}+4}$$

Divide 81+8\sqrt{2} by \sqrt{2}+4.

$$a=-\frac{7\sqrt{2}}{2}+22$$