All right, parts 1 and 2 complete. Let me know if you need any explanations!
Study Questions for Upcoming Exam
December 23, 2017
Graphically, we plot each equation as the equation of a line and find the intersection.
The lines intersect at the solution (x, y) = (−2, 1).
Algebraically, we solve using elimination. Multiplying through by 2 in our
first equation, we obtain
2y = x + 5
3x + 2y = 1
Then subtracting the first equation from the second, we obtain
3x = −x − 4 =⇒ x = −1
Then substituting in for x we have y = 21 (−1) + 52 = 2 so our solution is
(x, y) = (−1, 2) which agrees with the graphical solution.
Let t be the number of years from the initial starting time. Then the country’s
population in millions is modeled by p(t) = 2.5 + .8t while the annual food
supply is modeled by f (t) = 4 + .4t. Then there will first be food shortages
when f (t) = p(t). Solving, we have
2.5 + .8t = 4 + .4t
t = 3.75 years
The changes to the food supply can be represented as f2 (t) = 8 + .5t. Shortages
will still occur as the rate of increase in the population each year is greater than
the rate of increase in the food supply (so we know at some point the population
must overtake the food supply). We can find this point by setting p(t) = f2 (t)
and solving for t:
2.5 + .8t = 8 + .5t
t = 18 years
Therefore, a food shortage will still occur, but in 18 31 years, a much longer
We can represent our new food supply as f3 (t) = 8 + .8t. Then food shortages
would not occur because the food supply increases at the same rate as the
population and the food supply is initially greater than the population. That
is p(0) < f3 (0) and p(t) = f3 (t) has no solution:
2.5 + .8t = 8 + .8t
2.5 = 8
This equation clearly has no solution, so there will be no food shortages.
≤ x ≤ 600
≤ y ≤ 500
We graph only the intersecti...