# How do you find the value of y for a given value of x, if y varies directly with x. If y=166 when x=83, what is y when x=237?

May 16, 2017

$474$

#### Explanation:

Let's write this information in a table:

$\textcolor{w h i t e}{0} x \textcolor{w h i t e}{0} | \textcolor{w h i t e}{0} y \textcolor{w h i t e}{0}$
....................
$\textcolor{w h i t e}{0} 83 \textcolor{w h i t e}{} | \textcolor{w h i t e}{} 166 \textcolor{w h i t e}{0}$
color(white)()237 color(white)()|color(white)(0) ? color(white)(0)

Here's a pattern I noticed
$83 \times 2 = 166$

$y = 2 x$

Maybe the next pattern is
$237 \times 2 = 474$

May 16, 2017

$474$

#### Explanation:

$\text{given } y \propto x$

$\text{to convert to an equation, multiply by k, the constant}$
$\text{of variation}$

$\Rightarrow y = k \times x = k x$

$\text{to find k, use the given condition}$

$y = 166 \text{ when } x = 83$

$\Rightarrow k = \frac{y}{x} = \frac{166}{83} = 2$

$\text{the equation is } \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = 2 x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\Rightarrow x = 237 \to y = 2 \times 237 = 474$