A plan traveled for a total of 2,773 miles over the course of 8 hours. Heading south, the plan traveled at an average speed of 344 miles per hour and heading west. It traveled at an average speed of 351 miles per hour. For how many hours was the plane heading south?

  5 hours

Step-by-step explanation:

Let s and w represent hours spent traveling south and west, respectively. The problem statement tells us ...

  s + w = 8 . . . . . . total trip time

  344s + 351w = 2773 . . . . total mileage (using d = s·t)

By Cramer's rule, we can find the value of s as ...

  s = (1(2773) -351(8))/(1(344) -351(1)) = -35/-7 = 5

The plane was heading south for 5 hours.

Comment on Cramer's rule

For the equations ...

  ax +by = c

  dx +ey = f

The solution for x can be found to be ...

  x = (bf-ec)/(bd-ea)

Similarly, the solution for y is ...

  y = (cd-fa)/(bd-ea)

if gloria has "carry over" minutes, (that is, any minutes paid for but not used in a month can be applied to her payment for minutes in the next month,) her total usage for the three months is: total minutes = 320 + 243 + 489 = 1052 minutes number of 200 minute blocks = 1052/200 = 5 r52 (that is 5 with a remainder of 52) rounding up the number of 200 minute blocks gives her usage for the three months as 6 blocks. the cost for six blocks is 6 x $25 = $150 she also has 200 - 52 or 148 carry over minutes if gloria pays each month and unused minutes are lost, applying the rounding up procedure to each month gives: first month usage of 320 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 second month usage of 243 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 third month usage of 489 minutes rounds up to 3 blocks of 200 minutes for a payment of 3 x $25 = $75 total payments for the three months are $50 + $50 + $75 = $175 get the plan with carry over minutes!