The average rate of the first part of yi’s walk on a park loop was 4 miles per hour. she then met up with a friend and the two walked the rest of the way at an average rate of 5 miles per hour. the entire 3-mile walk took yi 42 minutes (0.7 hour). which equation can be used to solve for x, the time in hours that yi spent walking before meeting her friend?

So she walked during this time with 4 miles per hour (x*4)

during the rest of the way (which is 0.7-x, as the whole way took her 42 minutes, so the rest is 0.7-x) she walked with 5 miles per hour - the distance was (0..7-x)*5 m/h

the total distance was 3 miles, so if we sum the two distances, we will get 3 miles:

x*4+ (0.7-x)*5=3

let's remove the bracket:

4x+0.7*5-5x=3

4x+3.5-5x=3

subtract 3.5 from both sides:
4x-5x=3-3.5
-x=-0.5
 
multiply both sides by  (-1)
x=0.5:

so she walked for half an hour alone, that is for 30 minutes!