Question

(5 √3)/(5-√3)=x-√15y then calculate value of x y

Answer 1

THE ACTUAL QUESTION IS "(5+√3)/(5-√3)=x-√15y then calculate value of x+y".

SOLUTION:

The value of x+y is $\frac{14-\sqrt{5}}{11}$.

• It is given that (5+√3)/(5-√3)=x-√15y.
• We have to calculate the value of x and y.
• We solve the left hand side of the equation first.

LHS = $\frac{5 + \sqrt{3} }{5 - \sqrt{3}}$

• We rationalise the denominator by multiplying it with the conjugate of the denominator.

• The conjugate of 5-√3 is 5+√3.
• Multiplying the numerator and denominator with the conjugate, we get

LHS = $\frac{5 + \sqrt{3} }{5 - \sqrt{3}}$ × $\frac{5+\sqrt{3} }{5 + \sqrt{3}}$

LHS = $\frac{(5 + \sqrt{3})(5+\sqrt{3}) }{5^2 - (\sqrt{3})^2}$

LHS = $\frac{25 + 10\sqrt{3}+3 }{25 - 3}$

LHS = $\frac{28 + 10\sqrt{3} }{22}$

LHS = $\frac{5\sqrt{3}}{11} + \frac{14}{11}$

• Now comparing LHS and RHS we obtain,

$\frac{5\sqrt{3}}{11} + \frac{14}{11}$ = x - √15y

Now,

$\frac{\sqrt{5}\sqrt{15}}{11} + \frac{14}{11}$ = x - √15y

THerefore, x = $\frac{14}{11}$ and y = $\frac{-\sqrt{5}}{11}$

Now calculating x+y,

x + y = $\frac{14}{11}$ + $\frac{-\sqrt{5}}{11}$

x+y = $\frac{14 - \sqrt{5}}{11}$.