Question

Many people enjoy challenging themselves by running in marathons. A marathon is a 26-mile race. The runners wear numbers that identify them so their progress can be tracked at checkpoints throughout the race. Such races can be modeled using equations. An athlete is running in a marathon that eventually finishes at a local high school. After x minutes, the distance the athlete is from the school is given by y (in miles), where x +10y=260 note: 0

Answer 1

We can start by rewriting the initial equation x +10y=260 in order to express how y depends on x.
$10y=260-x$
$y=26-0.1x$
This is a linear function ( it's a straight line). Linear functions have two intercepts, y and x.
To find x-intercept (also called zero of a function) we set y=0 and we solve for x.
$0=26-0.1x$
$26=0.1x$
$x=260$
This means that at x=260 our function has a value of 0. y(x) represents a distance from the school. This means that when y(x) is 0 our athlete reached school (finished the race), so this also answers your second question. The time it takes an athlete to finish the race is 260 minutes.
The y-intercept a value of our function at x=0. This gives us a distance of our athlete from the school at the start of the race. We find this by plugin x=0 in our equation. We get that y=26 when x=0. This makes perfect sense. Our athlete is 26 miles away from high school at the start of the race.