Solve frac{10+asqrt{7}}{3+sqrt{7}}=b-2sqrt{7}

$$\frac { 10 + a \sqrt { 7 } } { 3 + \sqrt { 7 } } = b - 2 \sqrt { 7 }$$

Solve for a

$$a=\frac{3\sqrt{7}b}{7}+b-\frac{24\sqrt{7}}{7}-6$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{10+a\sqrt{7}}{3+\sqrt{7}} by multiplying numerator and denominator by 3-\sqrt{7}.

$$\frac{\left(10+a\sqrt{7}\right)\left(3-\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}=b-2\sqrt{7}$$

Consider \left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(10+a\sqrt{7}\right)\left(3-\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}=b-2\sqrt{7}$$

Square 3. Square \sqrt{7}.

$$\frac{\left(10+a\sqrt{7}\right)\left(3-\sqrt{7}\right)}{9-7}=b-2\sqrt{7}$$

Subtract 7 from 9 to get 2.

$$\frac{\left(10+a\sqrt{7}\right)\left(3-\sqrt{7}\right)}{2}=b-2\sqrt{7}$$

Use the distributive property to multiply 10+a\sqrt{7} by 3-\sqrt{7}.

$$\frac{30-10\sqrt{7}+3a\sqrt{7}-a\left(\sqrt{7}\right)^{2}}{2}=b-2\sqrt{7}$$

The square of \sqrt{7} is 7.

$$\frac{30-10\sqrt{7}+3a\sqrt{7}-a\times 7}{2}=b-2\sqrt{7}$$

Multiply -1 and 7 to get -7.

$$\frac{30-10\sqrt{7}+3a\sqrt{7}-7a}{2}=b-2\sqrt{7}$$

Multiply both sides of the equation by 2.

$$30-10\sqrt{7}+3a\sqrt{7}-7a=2b-4\sqrt{7}$$

Subtract 30 from both sides.

$$-10\sqrt{7}+3a\sqrt{7}-7a=2b-4\sqrt{7}-30$$

Add 10\sqrt{7} to both sides.

$$3a\sqrt{7}-7a=2b-4\sqrt{7}-30+10\sqrt{7}$$

Combine -4\sqrt{7} and 10\sqrt{7} to get 6\sqrt{7}.

$$3a\sqrt{7}-7a=2b+6\sqrt{7}-30$$

Combine all terms containing a.

$$\left(3\sqrt{7}-7\right)a=2b+6\sqrt{7}-30$$

Divide both sides by 3\sqrt{7}-7.

$$\frac{\left(3\sqrt{7}-7\right)a}{3\sqrt{7}-7}=\frac{2b+6\sqrt{7}-30}{3\sqrt{7}-7}$$

Dividing by 3\sqrt{7}-7 undoes the multiplication by 3\sqrt{7}-7.

$$a=\frac{2b+6\sqrt{7}-30}{3\sqrt{7}-7}$$

Divide 2b+6\sqrt{7}-30 by 3\sqrt{7}-7.

$$a=\frac{3\sqrt{7}b}{7}+b-\frac{24\sqrt{7}}{7}-6$$

Solve for b

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