Linear Problems
1. Find the equation of the line parallel to 3x+5y=6 and passing through the point (0,6).
2. Where do the lines y=3x+1 and 2y=x-7 cross?
3. Find where the line which passes through the two points (1,2) and (3,5) intersects the
line through the points (2,1) and (-1,1). Start by drawing a diagram.
4. Two car companies, Elvis and Hearts, rent cars. The Elvis car costs $30 plus 20 cents
per mile. The Hearts car costs $50 with 50 free miles, with extra mileage costing 15
cents per mile. On a single diagram, graph the cost against the distance driven for both
car companies.
a)
When is it cheaper to get an Elvis car?
b)
What is the significance of the slopes of the two graphs?
5. Suppose the points (3,5) and (10,17) are on line L.
a)
What point is midway between these points?
b)
What point is 1/3 of the way from (3,5) towards (10,17)?
c)
If (4,y) is on L, find y.
d)
If (x,8) is on L, find x.
6. In a meal plan at a college you pay a monthly membership fee, and then you receive
all meals at a predetermined fixed price per meal.
a)
If 30 meals cost $152.50 and 60 meals cost $250, find the membership fee and the
fixed price for a meal.
b)
Write a formula for the cost of a meal plan, C, in terms of the number of meal, m.
c)
Find the cost for 50 meals.
d)
Now find m in terms of C
e)
Used part d) to determine the maximum number of meals you can buy on a
budget of $300.
7. You need to rent a car and compare the charges of three different companies.
Company A charges 20 cents per mile plus a fixed rate of $20 per day. Company B
charges 10 cents per mile plus a fixed rate of $35 per day. Company C charges a flat rate
of $70 per day with no mileage charge.
a)
Find formulas for the cost of driving cars rented from companies A, B and C, in
terms of the x, the distance driven in miles in one day.
b)
Graph the costs for each company for 0<x<500. Put all three graphs on the same
set of axes.
c)
What do the slope and vertical intercept tell you in this situation?
d)
Use the graph in b) to find under what circumstances would company A be the
cheapest? Company B? Company C?
ANSWERS:
1. y = -(3/5)x+6
2. (-9/5 , -22/5)
3. (1/3 , 1)
4a. Elvis is cheaper if you drive less than 250 miles.
4b. Slopes tell you the cost per mile.
5a. (13/2 , 11)
5b. (16/3 , 9)
5c. (4 , 47/7)
5d. (19/4 , 8)
6a. Membership fee=$55; Fixed price per meal=$3.25
6b. C = 3.25m+55
6c. $217.50
6d. m = (C-55)/3.25
6e. 75 meals
7a. Company A: 0.2x+20
Company B: 0.1x+35
Company C: 70
7c. Slope = cost per mile
Vertical intercept = fixed cost per day
7d. A: x<150
B: 150<x<350
C: x>350