Let us call the food budget f, the entertainment budget e, and the rent budget r. We know the following about Luke's budget
[tex]f=3e[/tex][tex]r=3f[/tex]and
[tex]r+f+e=586.04[/tex]which can also be written as (since f =3e, its also true that e = f/3)
[tex]3f+f+\frac{f}{3}=586.04[/tex]we add all the f terms on the left to get
[tex](3+1+\frac{1}{3})f=586.04[/tex][tex](4+\frac{1}{3})f=586.04[/tex][tex](\frac{12}{3}+\frac{1}{3})f=586.04[/tex][tex]\frac{13f}{3}=586.04[/tex][tex]\textcolor{#FF7968}{\therefore f=135.24}[/tex]Hence, the maximum luke can spend on food is $135.24.
