MPM2DI
Lesson 1_5
Problems using Systems of Equations
1. At the deli, two smoked turkey subs and 5 veggie subs cost $29. Four smoked turkey subs and three veggie
subs cost $30.
a) Create a linear system to model the equations.
b) Solve the system to find the cost of a smoked turkey sub and the cost of a veggie sub.
2. The spa is offering two deals for new members. They can get 5 facials and 3 manicures for $128, or 2 facials
and 3 manicures for $62. What are the special prices of a facial and a manicure?
3. Michelle and Pierre both want to buy guitars. Michelle has already saved $60, and plans to save $5 per week
until she can buy the guitar. Pierre has $25, and plans to save $12 per week. How many weeks will pass
before Michelle and Pierre have the same amount of money?
4. “Le Sage” Restaurant charges one price for adults and another for children for their buffet dinner. Raj’s
family has two adults and three children and their bill was $57 without taxes. Eman’s family consists of 3
adults and 2 children and their bill was $63 without taxes. What is the cost of the buffet for an adult and for a
child.
5. Supplementary angles are angles that have a sum of
is
greater than
, what are the values of
and
. If angles
?
and
are supplementary, and
6. By the end of 2006, The Red Rock Pirates’ high school hockey team and the Blue Haven Lions’ high school
hockey team had won a total of 31 hockey championships between them. The Pirates have won 5.2 times as
many championships as the Lions. How many hockey championships have each team won?
7. The sum of two numbers is 66 and their difference is 18. What are the numbers?
8. Three times one number added to another number is 39. Twice the first number minus the other is 6. What are
the numbers?
9. Solve the following linear system.
10. Solve the following linear system.
11. One metal alloy is 25% copper, while another is 50% copper. How much of each alloy should be used to
make 1500 g of a metal alloy that is 40% copper?
12. Chris needs to make 500 L of a 35% acidic solution. He has only two of the acidic solutions available, a 25%
solution and a 50% solution. How many litres of each acidic solution should he mix?
13. Raylene is making a healthy snack of trail mix to sell as part of a fundraiser for the school hiking club. The
mixture will have 2.5 times as many kilograms of raisins as sunflower seeds. Sunflower seeds cost $12/kg and
raisins cost $6/kg. If Raylene has $108 to spend, what mass of sunflower seeds and of raisins should she buy?
14. A houseboat on the Trent river system travelled 48 km upstream in 6 h. It only took the houseboat 4 h to
make the same trip downstream. Some of the time the water was still, some of the time there was a current.
(a) How fast would the houseboat have travelled in still water?
(b) How fast was the river’s current?
15. A salmon fishing boat on a BC river travelled upstream 72 km in 4 h. Returning downstream at the same
speed, it took 3 h.
(a) Find the speed of the fishing boat in still water.
(b) Find the speed of the river’s current.
16. The school band bought tickets for a concert. They paid $290 for 6 tickets in Section A and 10 tickets in
Section B. When the concert was repeated the next week, they paid $220 for 4 tickets in Section A and 8
tickets in Section B.
(a) Write a linear system to model this problem.
(b) What are the prices of the tickets in Sections A and B?
17. The cost of admission to Fantasy World theme park totalled $120.50 for a group of 11 children and 2 adults.
The admission totalled $100 for another group consisting of 7 children and 3 adults.
(a) Write a linear system to model this problem.
(b) What is the admission cost for an adult and for a child?
18. On Canada Day three families went to a theme park for the day. The first family had 2 senior citizens, 2
children, and 1 adult, and paid $57. The second family had 1 senior citizen, 3 children, and 2 adults, and paid
$69. The third family had 2 senior citizens, 3 children, and 3 adults, and paid $96.
(a) Write a linear system to model this problem.
(b) Find the cost of admission for a senior citizen, an adult and a child.
19. Frank used 45 min of peak time and 50 min of non-peak time on his cell phone last month. His bill was
$37.25. His friend Gordon used 70 min of peak time and 30 min of non-peak time on his cell phone in the
same month. His bill was $46.
(a) Write a linear system to model the problem.
(b) What were the peak and non peak rates?
20. An airliner travelling from Toronto to Vancouver took 5h to cover the 3900 km trip against a headwind. The
return trip, travelling with a tailwind that was twice the speed of the headwind took 4h 20 min. How fast were
the headwind and tailwind on the two trips? How fast would the airliner have flown in still air?
21. Candace’s grandmother invested $5000 for her future college education. She placed part of the money in an
account that paid 4% simple interest each year, and the rest in a government bond that paid 3.5% simple
interest each year. After one year her investments had earned a total of $190.00 in interest. How much money
did she put into each investment?
Answers:
1.
a) Let the cost of a smoked turkey sub in dollars be t and the cost of a veggie sub in dollars be v.
...(1)
...(2)
b) A smoked turkey sub costs $4.50 and a veggie sub costs $4.
2. Let represent the price of a facial and the price of a manicure, both in dollars.
The price for a facial is $22, and the price for a manicure is $6.
3. After saving for w weeks Michelle will have
If they have the same amount, then
the same amount of money.
dollars, and Pierre will have
dollars.
. After 5 weeks, Michelle and Pierre will have
4. Let the cost for an adult be dollars, and the cost for a child be dollars.
...(1)
...(2)
The cost is $15 for an adult and $9 for a child.
5.
...(1)
...(2)
The value of
is
, and the value of
is
.
6. Let p represent the number of championships the Pirates have won, and l represent the number of
championships the Lions have won.
...(1)
...(2)
The Lions have won 5 championships, and the Pirates have won 26 championships.
7. Let n and m represent the two numbers.
...(1)
...(2)
Add equation (1) and equation (2):
The two numbers are 42 and 24.
8. Let x and y represent the numbers.
...(1)
...(2)
Add equation (1) and equation (2):
The two numbers are 9 and 12.
9.
Express x in terms of y in the second equation, then substitute that value in the first equation. Solve for y,
and then substitute that value back into the second equation to find the value of x.
10.
Express x in terms of y in the first equation, then substitute that value in the second equation. Solve for y,
and then substitute that value back into the first equation to find the value of x.
11. Let represent the amount of the 25% copper alloy used, and y represent the amount of the 50%
alloy used.
...(1)
...(2)
Rearrange equation (1):
Substitute
in equation (2):
Substitute
y
in rearranged equation (1):
To make 1500 g of an alloy that is 40% copper, 600 g of the 25% copper alloy and 900g of the 50%
copper alloy should be used.
12. Let litres represent the number of litres of the 25% acidic solution to use, and
number of litres of the 50% acidic solution to use.
...(1)
...(2)
Rearrange equation (1):
Substitute
Substitute
y
represent the
in equation (2):
in rearranged equation (1):
To make the 35% acidic solution, Chris should mix 300 L of the 25% solution and 200 L of the 50%
solution.
13. Let x be the mass of sunflower seeds and y the mass of raisins in Raylene’s trail mix.
...(1)
...(2)
Raylene should buy 4 kg of sunflower seeds and 10 kg of raisins.
14. Let the speed of the houseboat in still water be h, and the speed of the river’s current be c, both in
kilometres per hour.
Upstream:
...(1)
Downstream:
...(2)
Simplify equation (1):
Simplify equation (2):
Add simplified equation (1) to simplified equation (2):
Substitute
in simplified equation (2):
The houseboat travelled at 10 km/h in still water, and the river current was 2 km/h.
15. Let f be the speed of the fishing boat in still water, and c be the speed of the river’s current.
Upstream:
...(1)
Downstream:
...(2)
Simplify equation (1):
Simplify equation (2):
Add simplified equation (1) and simplified equation (2):
Substitute
in simplified equation (2):
The fishing boat’s speed in still water was 21 km/h, and the river’s current was 3 km/h.
16.
(a) Let represent the cost of a ticket in Section A, and y represent the cost of ticket in Section B.
...(1)
...(2)
(b) Multiply equation (1) by 2, and equation (2) by 3, then subtract them:
Substitute
in equation (1):
The price of a ticket in Section A is $15, and the price of a ticket in Section B is $20.
17.
(a) Let the admissions cost, in dollars, for a child be c and for an adult be .
...(1)
...(2)
(b) Multiply equation (1) by 3, and equation (2) by 2, then subtract them.
Substitute
in equation (2):
The cost of admission for an adult is $13.50 and the cost of admission for a child is $8.50.
18.
(a) Let the costs of admission, in dollars, for a senior citizen be s, for an adult be a, and for a child be c.
...(1)
...(2)
...(3)
b) Rearrange equation (1):
Substitute
in equation (2):
Substitute
...(4)
in equation (3):
...(5)
Multiply equation (4) by 3, and add to equation (5):
Substitute
in equation (4):
Substitute
a
and
in rearranged equation (1):
The admission fees for a senior citizen, an adult, and a child are $12, $15, and $9, respectively.
19.
(a) Let the cost of peak time in dollars per minute be , and the cost of of non-peak time in dollars per
minute be .
...(1)
...(2)
(b)Multiply equation (1) by 3 and equation (2) by 5, and then subtract them:
Substitute
in equation (2):
The peak time charge was $0.55/min, and the non-peak charge was $0.25/min.
20. Let the speed of the airliner in still air be x, and the speed of the headwind be y, both in km/h. The
tailwind is twice the speed of the headwind (2y).
Toronto to Vancouver
...(1)
On the return trip:
...(2)
Rearrange equation (1):
Rearrange equation (2):
Subtract rearranged equation (1) and rearranged equation (2):
Substitute
in rearranged equation (1):
The headwind speed on the Toronto to Vancouver trip was 40 km/h and the speed of the tailwind on the
return trip was 80 km/h . The speed of the airliner in still air was 820 km/h .
21. Let x represent the number of dollars invested at 4%, and y represent the number of dollars invested
at 3.5%.
...(1)
...(2)
Rearrange equation (1):
Substitute
Substitute
y
in equation (2):
in rearranged equation (1):
Candace’s grandmother invested $3000 at 4% interest, and $2000 at 3.5% interest.