Solve frac{sqrt{2}}{2}+frac{sqrt{7}}{3}+frac{sqrt{13}}{5}x=1+2+3

$$\frac { \sqrt { 2 } } { 2 } + \frac { \sqrt { 7 } } { 3 } + \frac { \sqrt { 13 } } { 5 } x = 1 + 2 + 3$$

Solve for x

$$x=\frac{30\sqrt{13}}{13}-\frac{5\sqrt{26}}{26}-\frac{5\sqrt{91}}{39}\approx 6.116923292$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 2,3,5.

$$15\sqrt{2}+10\sqrt{7}+6\sqrt{13}x=30+60+90$$

Add 30 and 60 to get 90.

$$15\sqrt{2}+10\sqrt{7}+6\sqrt{13}x=90+90$$

Add 90 and 90 to get 180.

$$15\sqrt{2}+10\sqrt{7}+6\sqrt{13}x=180$$

Subtract 15\sqrt{2} from both sides.

$$10\sqrt{7}+6\sqrt{13}x=180-15\sqrt{2}$$

Subtract 10\sqrt{7} from both sides.

$$6\sqrt{13}x=180-15\sqrt{2}-10\sqrt{7}$$

The equation is in standard form.

$$6\sqrt{13}x=180-10\sqrt{7}-15\sqrt{2}$$

Divide both sides by 6\sqrt{13}.

$$\frac{6\sqrt{13}x}{6\sqrt{13}}=\frac{180-10\sqrt{7}-15\sqrt{2}}{6\sqrt{13}}$$

Dividing by 6\sqrt{13} undoes the multiplication by 6\sqrt{13}.

$$x=\frac{180-10\sqrt{7}-15\sqrt{2}}{6\sqrt{13}}$$

Divide 180-15\sqrt{2}-10\sqrt{7} by 6\sqrt{13}.

$$x=\frac{5\sqrt{13}\left(36-2\sqrt{7}-3\sqrt{2}\right)}{78}$$