As part of her training routine for basketball, Shaylle alternates between cycling and running for exercise. She cycles at a rate of 14 mph and runs at a rate of 8 mph. If she spends 4.5 hours exercising and covers a total of 51 miles, how much time did she spend on each exercise?
c + r = 4.5
14c + 8r = 51
solve by substitution or elimination
To solve this problem, we can set up a system of equations. Let's call the time Shaylle spent cycling "x" hours, and the time she spent running "y" hours.
From the problem, we know two things:
1. Shaylle cycles at a rate of 14 mph, so the distance she covers while cycling is 14x miles.
2. Shaylle runs at a rate of 8 mph, so the distance she covers while running is 8y miles.
According to the problem, Shaylle exercises for a total of 4.5 hours and covers a total distance of 51 miles. So, we can write two equations based on the information above:
Equation 1: x + y = 4.5 (because she spends a total of 4.5 hours exercising)
Equation 2: 14x + 8y = 51 (because the total distance covered is 51 miles)
To solve this system of equations, we can use substitution or elimination. Let's solve it using the substitution method:
From Equation 1, we can express x in terms of y by subtracting y from both sides:
x = 4.5 - y
Now, substitute this value of x into Equation 2:
14(4.5 - y) + 8y = 51
Distribute 14:
63 - 14y + 8y = 51
Combine like terms:
-6y = -12
Divide both sides by -6:
y = 2
Now, substitute this value of y back into Equation 1 to find x:
x + 2 = 4.5
x = 4.5 - 2
x = 2.5
So, Shaylle spent 2.5 hours cycling and 2 hours running.