# How do you solve sqrt((5y)/6)-10=4 and check your solution?

May 31, 2017

See a solution process below:

#### Explanation:

First, add $\textcolor{red}{10}$ to each side of the equation to isolate the radical while keeping the equation balanced:

$\sqrt{\frac{5 y}{6}} - 10 + \textcolor{red}{10} = 4 + \textcolor{red}{10}$

$\sqrt{\frac{5 y}{6}} - 0 = 14$

$\sqrt{\frac{5 y}{6}} = 14$

Next, square each side of the equation to eliminate the radical while keeping the equation balanced:

${\left(\sqrt{\frac{5 y}{6}}\right)}^{2} = {14}^{2}$

$\frac{5 y}{6} = 196$

Now, multiply each side of the equation by $\frac{\textcolor{red}{6}}{\textcolor{b l u e}{5}}$ to solve for $y$ while keeping the equation balanced:

$\frac{\textcolor{red}{6}}{\textcolor{b l u e}{5}} \times \frac{5 y}{6} = \frac{\textcolor{red}{6}}{\textcolor{b l u e}{5}} \times 196$

$\frac{\cancel{\textcolor{red}{6}}}{\cancel{\textcolor{b l u e}{5}}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{5}}} y}{\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}}} = \frac{1176}{5}$

$y = \frac{1176}{5}$

To check the solution we need to substitute $\textcolor{red}{\frac{1176}{5}}$ for $\textcolor{red}{y}$, calculate each side of the equation and ensure the two results are equal:

$\sqrt{\frac{5 \textcolor{red}{y}}{6}} - 10 = 4$ becomes:

$\sqrt{\frac{5 \times \textcolor{red}{\frac{1176}{5}}}{6}} - 10 = 4$

$\sqrt{\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} \times \textcolor{red}{\frac{1176}{\textcolor{b l a c k}{\cancel{\textcolor{red}{5}}}}}}{6}} - 10 = 4$

$\sqrt{\frac{1176}{6}} - 10 = 4$

$\pm \sqrt{196} - 10 = 4$

$\pm 14 - 10 = 4$

$4 = 4$ or $- 24 = 4$

The $- 14$ result of the square root is extraneous.

Therefore, $4 = 4$ and the solution is shown to be correct.

May 31, 2017

$y = 235.2$

#### Explanation:

$\sqrt{\frac{5 y}{6}} - 10 = 4$

$\sqrt{\frac{5 y}{6}} = 4 + 10$

$\sqrt{\frac{5 y}{6}} = 14$

$\frac{5 y}{6} = {14}^{2}$

$\frac{5 y}{6} = 196$

$5 y = 196 \times 6$

$5 y = 1176$

y = 1176 ÷ 5

color(blue)(y = 235.2

We can now substitute $y$ for $235.2$ to prove our answer.

$\sqrt{\frac{5 y}{6}} - 10 = 4$

$\sqrt{\frac{5 \times 235.2}{6}} - 10 = 4$

$\sqrt{\frac{1176}{6}} - 10 = 4$

$\sqrt{196} - 10 = 4$

$14 - 10 = 4$