Let:
x = Length of the short loop
y = Length of the long loop
Joel ran 3 short loops and 4 long loops, the total distance was 30mi, so:
[tex]3x+4y=30_{\text{ }}(1)[/tex]He also ran 2 short loops and 6 long loops, the total distance was 40mi, so:
[tex]2x+6y=40_{\text{ }}(2)[/tex]From (1) solve for x:
[tex]\begin{gathered} 3x=30-4y \\ x=10-\frac{4}{3}y_{\text{ }}(3) \end{gathered}[/tex]Replace (3) into (2):
[tex]\begin{gathered} 2(10-\frac{4}{3}y)+6y=40 \\ 20-\frac{8}{3}y+6y=40 \\ \frac{10}{3}y=20 \\ y=\frac{60}{10} \\ y=6 \end{gathered}[/tex]Replace y into (3):
[tex]\begin{gathered} x=10-\frac{4}{3}(6) \\ x=2 \end{gathered}[/tex]Answer:
The short loop has a length of 2 miles
The long loop has a length of 6 miles
