Question

t a gas station, 15 gallons of gas has a price of $17.55 and 18 gallons of gas has a price of $21.06. Write a direct variation equation that relates the cost y of any number of gallons of gas x. How many gallons of gas can be purchased for $23.40?

Answer 1

Use  constant of variation y =kx  i.e
15 gallons k=>17.55
k gallons =17.55/15=1.17

18 gallons k=>21.06
k gallons =21.06/18=1.17

Therefore ans => y=1.17x

Answer 2

A direct variation means, the value of the dependent variable,
varies based on the independent variable times some constant,
let us call it "k", so y=kx, whatever k is
so $\begin{array}{cccccclllll} \textit{something}&&\textit{varies directly to}&&\textit{something else}\\ \quad \\ \textit{something}&=&{{ \textit{some value}}}&\cdot &\textit{something else}\\ \quad \\ y&=&{{ k}}&\cdot&x \\ \quad \\ && y={{ (k) }}x \end{array}$
$\textit{we know for 15 gallons(x), the price(y) is 17.55} \begin{cases} x=15\\ y=17.55 \end{cases} \\ \quad \\ y=(k)x\implies 17.55=(k)15 \\ \quad \\ \textit{we know for 18 gallons(x), the price(y) is 21.06} \begin{cases} x=18\\ y=21.06 \end{cases} \\ \quad \\ y=(k)x\implies 21.06=(k)18 \\ \quad \\ \textit{what is "k"? or constant of variation, solve either above for "k"}$

once you get what "k" is, plug it back in the original y=(k)x

then, how many gallons can be purchased with 23.40? namely
if y=23.40, what's "x"?   put the "k" value found and solve for "x"