Cabrina and Dabney are at a conference in Seattle. Cabrina drives home to Boise at an average speed of 65 mph and Dabney drives home to Portland at 50 mph. If the sum of their driving times is 11.4 hours and if the sum of the distances driven is 690 miles, determine the time each person spent driving.

Cabrina's distance --- x miles

Dabney's distance = 690-x

Cabrina's time = x/65
Dabneys time = (690-x)/50

x/65 + (690-x)/50 = 11.4
times 650 , the LCD
10x + 13(690-x) = 7410
10x + 8970 - 13x = 7410
-3x = -1560
x = 520

continue

To solve this problem, we'll use the formula Distance = Speed * Time.

Let's assume that Cabrina drove for x hours and Dabney drove for y hours.

Since Cabrina drove at a speed of 65 mph and Dabney drove at a speed of 50 mph, we can set up the equations:

65x + 50y = 690 (Equation 1, representing the sum of the distances driven)
x + y = 11.4 (Equation 2, representing the sum of the driving times)

We now have a system of two equations with two unknowns. We can solve this system to find the values of x and y.

First, let's solve Equation 2 for x:
x = 11.4 - y

Now, substitute this equation into Equation 1:
65(11.4 - y) + 50y = 690

Simplifying this equation:
741 - 65y + 50y = 690
741 - 15y = 690
-15y = -51
y = 3.4

Now that we have the value of y, we can substitute it back into Equation 2 to find x:
x + 3.4 = 11.4
x = 11.4 - 3.4
x = 8

Therefore, Cabrina spent 8 hours driving and Dabney spent 3.4 hours driving.