Find y, if \( \sqrt{12} - \sqrt{147} + y\sqrt{3} = 0 \)

Answer Details

We are given an equation: $$ \sqrt{12} - \sqrt{147} + y\sqrt{3} = 0 $$ To solve for $y$, we can isolate the $\sqrt{3}$ term on one side of the equation: \begin{align*} \sqrt{12} - \sqrt{147} + y\sqrt{3} &= 0 \\ y\sqrt{3} &= \sqrt{147} - \sqrt{12} \\ y &= \frac{\sqrt{147} - \sqrt{12}}{\sqrt{3}} \\ \end{align*} To simplify the expression, we can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$: \begin{align*} y &= \frac{(\sqrt{147} - \sqrt{12})\sqrt{3}}{\sqrt{3}\sqrt{3}} \\ y &= \frac{\sqrt{441} - \sqrt{36}}{3} \\ y &= \frac{21 - 6}{3} \\ y &= \boxed{5} \end{align*} Therefore, $y=5$.