If y varies directly with x then
[tex]\begin{gathered} y=kx \\ \text{ Where k is a constant of a variation} \end{gathered}[/tex]First, we need to find the constant of a variation k, for this, we use the given values of x and y:
[tex]\begin{gathered} y=kx \\ 12=k\cdot15 \\ \text{ Divide by 15 from both sides of the equation} \\ \frac{12}{15}=\frac{k\cdot15}{15} \\ \frac{12}{15}=k \\ \text{ Simplifying} \\ \frac{3\cdot4}{3\cdot5}=k \\ \frac{4}{5}=k \end{gathered}[/tex]Then since we already have the value of k we can find the value x when y = 21:
[tex]\begin{gathered} y=kx \\ 21=\frac{4}{5}x \\ \text{ Multiply by 5 from both sides of the equation} \\ 5\cdot21=5\cdot\frac{4}{5}x \\ 105=4x \\ \text{ Divide by 4 from both sides of the equation} \\ \frac{105}{4}=\frac{4x}{4} \\ \frac{105}{4}=x \end{gathered}[/tex]Therefore, if y = 12 when x = 15, then
[tex]x=\frac{105}{4}[/tex]when y = 21.
