One competitor in a 100-mile bicycle race took a total of 5 hours to complete the course. his average speed in the morning was 23 miles per hour. his average speed in the afternoon was 13 miles per hour. how many hours did he ride in the morning, and how many hours did he ride in the afternoon?

Let x = morning ride time, y = afternoon ride time.

We set up two different equations to represent the given info in the description. The first one is morning time + afternoon time = total time to complete the course. So it's x + y = 5. Next, we do morning average multiplied by morning ride time + afternoon average multiplied by afternoon ride time = total distance. So it's 23x + 13y = 100. Then, set up a system of equations:

x + y = 5
23x + 13y = 100

Now, we must pick which variable we want to solve for first. This way we can plug the number we solve for back into one of the original equations to find the other missing variable. I'm going to solve for x first so we multiply by the first equation by - 13. You'll see why in a bit. 

- 13(x + y = 5) >> - 13x - 13y = - 65

Your new system of equations is:

- 13x - 13y = - 65
23x + 13y = 100

Now we combine like terms amongst the two equations. Notice the y-variables will cancel each other out, leaving us with just the x to solve for like I want. 

10x = 35

Divide by 10 on both sides to isolate variable x

x = 35/10 or 3.5

Now that we have x, we solve for y. Plug your x-value back into either equation; I choose the first because it's easiest. 

3.5 + y = 5

Subtract 3.5 from both sides to isolate variable y.

3.5 + y - 3.5 = 5 - 3.5
y + 0 = 1.5
y = 1.5

So the competitor rode 3.5 hours in the morning and 1.5 hours in the afternoon. 

OR

The competitor rode 3 hours, 30 minutes in the morning and 1 hour, 30 minutes in the afternoon.