Solve 76/4+sqrt{7+sqrt{11}}=p+qsqrt{7}+rsqrt{11}+ssqrt{77}

$$\frac { 76 } { 4 + \sqrt { 7 } + \sqrt { 11 } } = p + q \sqrt { 7 } + r \sqrt { 11 } + s \sqrt { 77 }$$

Solve for q

$$q=\frac{\left(14-3\sqrt{7}\right)\left(-2\sqrt{7}p-2\sqrt{77}r-3\sqrt{11}r-3\sqrt{77}s-14\sqrt{11}s-3p+19\sqrt{7}+76-19\sqrt{11}\right)}{133}$$

Steps for Solving Linear Equation

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

$$p+q\sqrt{7}+r\sqrt{11}+s\sqrt{77}=\frac{76}{4+\sqrt{7}+\sqrt{11}}$$

Subtract p from both sides.

$$q\sqrt{7}+r\sqrt{11}+s\sqrt{77}=\frac{76}{4+\sqrt{7}+\sqrt{11}}-p$$

To add or subtract expressions, expand them to make their denominators the same. Multiply p times \frac{4+\sqrt{7}+\sqrt{11}}{4+\sqrt{7}+\sqrt{11}}.

$$q\sqrt{7}+r\sqrt{11}+s\sqrt{77}=\frac{76}{4+\sqrt{7}+\sqrt{11}}-\frac{p\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}}$$

Since \frac{76}{4+\sqrt{7}+\sqrt{11}} and \frac{p\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}} have the same denominator, subtract them by subtracting their numerators.

$$q\sqrt{7}+r\sqrt{11}+s\sqrt{77}=\frac{76-p\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}}$$

Do the multiplications in 76-p\left(4+\sqrt{7}+\sqrt{11}\right).

$$q\sqrt{7}+r\sqrt{11}+s\sqrt{77}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}}{4+\sqrt{7}+\sqrt{11}}$$

Subtract r\sqrt{11} from both sides.

$$q\sqrt{7}+s\sqrt{77}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}}{4+\sqrt{7}+\sqrt{11}}-r\sqrt{11}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply r\sqrt{11} times \frac{4+\sqrt{7}+\sqrt{11}}{4+\sqrt{7}+\sqrt{11}}.

$$q\sqrt{7}+s\sqrt{77}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}}{4+\sqrt{7}+\sqrt{11}}-\frac{r\sqrt{11}\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}}$$

Since \frac{76-4p-p\sqrt{7}-p\sqrt{11}}{4+\sqrt{7}+\sqrt{11}} and \frac{r\sqrt{11}\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}} have the same denominator, subtract them by subtracting their numerators.

$$q\sqrt{7}+s\sqrt{77}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}-r\sqrt{11}\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}}$$

Do the multiplications in 76-4p-p\sqrt{7}-p\sqrt{11}-r\sqrt{11}\left(4+\sqrt{7}+\sqrt{11}\right).

$$q\sqrt{7}+s\sqrt{77}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r}{4+\sqrt{7}+\sqrt{11}}$$

Subtract s\sqrt{77} from both sides.

$$q\sqrt{7}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r}{4+\sqrt{7}+\sqrt{11}}-s\sqrt{77}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply s\sqrt{77} times \frac{4+\sqrt{7}+\sqrt{11}}{4+\sqrt{7}+\sqrt{11}}.

$$q\sqrt{7}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r}{4+\sqrt{7}+\sqrt{11}}-\frac{s\sqrt{77}\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}}$$

Since \frac{76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r}{4+\sqrt{7}+\sqrt{11}} and \frac{s\sqrt{77}\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}} have the same denominator, subtract them by subtracting their numerators.

$$q\sqrt{7}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r-s\sqrt{77}\left(4+\sqrt{7}+\sqrt{11}\right)}{4+\sqrt{7}+\sqrt{11}}$$

Do the multiplications in 76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r-s\sqrt{77}\left(4+\sqrt{7}+\sqrt{11}\right).

$$q\sqrt{7}=\frac{76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r-4s\sqrt{77}-7s\sqrt{11}-11s\sqrt{7}}{4+\sqrt{7}+\sqrt{11}}$$

The equation is in standard form.

$$\sqrt{7}q=\frac{-4\sqrt{11}r-4\sqrt{77}s-7\sqrt{11}s-11\sqrt{7}s-\sqrt{7}p-\sqrt{11}p-\sqrt{77}r-4p-11r+76}{\sqrt{7}+\sqrt{11}+4}$$

Divide both sides by \sqrt{7}.

$$\frac{\sqrt{7}q}{\sqrt{7}}=\frac{\left(3\sqrt{11}+5\sqrt{7}+2-2\sqrt{77}\right)\left(-4\sqrt{11}r-4\sqrt{77}s-7\sqrt{11}s-11\sqrt{7}s-\sqrt{7}p-\sqrt{11}p-\sqrt{77}r-4p-11r+76\right)}{76\sqrt{7}}$$

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

$$q=\frac{\left(3\sqrt{11}+5\sqrt{7}+2-2\sqrt{77}\right)\left(-4\sqrt{11}r-4\sqrt{77}s-7\sqrt{11}s-11\sqrt{7}s-\sqrt{7}p-\sqrt{11}p-\sqrt{77}r-4p-11r+76\right)}{76\sqrt{7}}$$

Divide \frac{\left(-2\sqrt{77}+5\sqrt{7}+3\sqrt{11}+2\right)\left(76-4p-p\sqrt{7}-p\sqrt{11}-4r\sqrt{11}-r\sqrt{77}-11r-4s\sqrt{77}-7s\sqrt{11}-11s\sqrt{7}\right)}{76} by \sqrt{7}.

$$q=\frac{\sqrt{7}\left(3\sqrt{11}+5\sqrt{7}+2-2\sqrt{77}\right)\left(-4\sqrt{11}r-4\sqrt{77}s-7\sqrt{11}s-11\sqrt{7}s-\sqrt{7}p-\sqrt{11}p-\sqrt{77}r-4p-11r+76\right)}{532}$$

Solve for p

$$p=-\frac{\left(3-2\sqrt{7}\right)\left(-2\sqrt{77}r-3\sqrt{7}q-3\sqrt{11}r-3\sqrt{77}s-14\sqrt{11}s-14q+19\sqrt{7}+76-19\sqrt{11}\right)}{19}$$