we know that
A relationship between two variables, x, and y, represent a direct variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]
in this problem we have
the point [tex](5,12)[/tex] is on the line of direct variation
so
Find the constant of proportionality k
[tex]y/x=k[/tex]-------> substitute ------> [tex]k=12/5[/tex]
the equation is
[tex]y=\frac{12}{5}x[/tex]
Remember that
If a point is on the line of direct variation
then
the point must satisfy the equation of direct variation
we're proceeding to verify each point
case A) point [tex](0,0)[/tex]
[tex]x=0\ y=0[/tex]
Substitute the value of x and y in the direct variation equation
[tex]0=\frac{12}{5}*0[/tex]
[tex]0=0[/tex] -------> is true
therefore
the point [tex](0,0)[/tex] is on the line of direct variation
case B) point [tex](2.5,6)[/tex]
[tex]x=2.5\ y=6[/tex]
Substitute the value of x and y in the direct variation equation
[tex]6=\frac{12}{5}*2.5[/tex]
[tex]6=6[/tex] -------> is true
therefore
the point [tex](2.5,6)[/tex] is on the line of direct variation
case C) point [tex](3,10)[/tex]
[tex]x=3\ y=10[/tex]
Substitute the value of x and y in the direct variation equation
[tex]10=\frac{12}{5}*3[/tex]
[tex]10=7.2[/tex] -------> is not true
therefore
the point [tex](3,10)[/tex] is not on the line of direct variation
case D) point [tex](7.5,18)[/tex]
[tex]x=7.5\ y=18[/tex]
Substitute the value of x and y in the direct variation equation
[tex]18=\frac{12}{5}*7.5[/tex]
[tex]18=18[/tex] -------> is true
therefore
the point [tex](7.5,18)[/tex] is on the line of direct variation
case E) point [tex](12.5,24)[/tex]
[tex]x=12.5\ y=24[/tex]
Substitute the value of x and y in the direct variation equation
[tex]24=\frac{12}{5}*12.5[/tex]
[tex]18=30[/tex] -------> is not true
therefore
the point [tex](12.5,24)[/tex] is not on the line of direct variation
case F) point [tex](15,36)[/tex]
[tex]x=15\ y=36[/tex]
Substitute the value of x and y in the direct variation equation
[tex]36=\frac{12}{5}*15[/tex]
[tex]36=36[/tex] -------> is true
therefore
the point [tex](15,36)[/tex] is on the line of direct variation
therefore
the answer is
[tex](0,0)[/tex]
[tex](2.5,6)[/tex]
[tex](7.5,18)[/tex]
[tex](15,36)[/tex]
