Solve 1/2*2p/sqrt{3}+2p=50/sqrt{2}

$$\frac { 1 } { 2 } \times \frac { 2 p } { \sqrt { 3 } } + 2 p = \frac { 50 } { \sqrt { 2 } }$$

Solve for p

$$p = \frac{25 \sqrt{6} {(2 \sqrt{3} - 1)}}{11} \approx 13.717708253$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{2p}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{1}{2}\times \frac{2p\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+2p=\frac{50}{\sqrt{2}}$$

The square of \sqrt{3} is 3.

$$\frac{1}{2}\times \frac{2p\sqrt{3}}{3}+2p=\frac{50}{\sqrt{2}}$$

Multiply \frac{1}{2} times \frac{2p\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2p\sqrt{3}}{2\times 3}+2p=\frac{50}{\sqrt{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{3}p}{3}+2p=\frac{50}{\sqrt{2}}$$

Rationalize the denominator of \frac{50}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{3}p}{3}+2p=\frac{50\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}p}{3}+2p=\frac{50\sqrt{2}}{2}$$

Divide 50\sqrt{2} by 2 to get 25\sqrt{2}.

$$\frac{\sqrt{3}p}{3}+2p=25\sqrt{2}$$

Multiply both sides of the equation by 3.

$$\sqrt{3}p+6p=75\sqrt{2}$$

Combine all terms containing p.

$$\left(\sqrt{3}+6\right)p=75\sqrt{2}$$

Divide both sides by \sqrt{3}+6.

$$\frac{\left(\sqrt{3}+6\right)p}{\sqrt{3}+6}=\frac{75\sqrt{2}}{\sqrt{3}+6}$$

Dividing by \sqrt{3}+6 undoes the multiplication by \sqrt{3}+6.

$$p=\frac{75\sqrt{2}}{\sqrt{3}+6}$$

Divide 75\sqrt{2} by \sqrt{3}+6.

$$p=-\frac{25\sqrt{2}\left(\sqrt{3}-6\right)}{11}$$