Chapter 4.1, Problem 79SR

To determine

To Calculate:

If y = 12 when x = 15, find x when y = -6.

Answer to Problem 79SR

x = -7.5 , when y = -6

Explanation of Solution

Given Information:

If y = 12 when x = 15 .

Concept Used:

Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value. It is represented by the proportional symbol , $\alpha$ .

For example, x and y are two quantities or variables which are linked with each other directly, then we can say $x\alpha y$ . When we remove the proportionality symbol, the ratio of x and y becomes equal to a constant, such as $\frac{x}{y}=C$ , where C is a constant. 

Calculation:

Since y varies directly as x .

  $\begin{array}{l}y=kx\\ 12=k(15)\mathrm{.....................}["y=12"when"x=15"]\\ \frac{12}{15}=\frac{k(15)}{15}\mathrm{...................}[divide"15"each\_side]\\ \frac{4}{5}=k\mathrm{...........................}[simplify]\end{array}$

So, the direct variation equation is $y=\frac{4}{5}x$ .

Substitute y = -6 and find x :

  $\begin{array}{l}y=\frac{4}{5}x\\ -6=\frac{4}{5}x\mathrm{.....................}[y=-6]\\ -6(\frac{5}{4})=\frac{4x}{5}(\frac{5}{4})\mathrm{...................}[divide"\frac{5}{4}"each\_side]\\ -\frac{15}{2}=x\mathrm{..............................}[simplify]\\ x=-7.5\end{array}$

So, x = -7.5 when y = -6.