Problem Solving

1
Chapter 1 Systems of Equations
1.3 and 1.5
Problem Solving
1.1
MATHPOWERTM 11, WESTERN EDITION
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Problem 1 The sum of two numbers is 16. Twice
the first number plus three times the second
number is 34. Find the two numbers.
2x 2y 32 2x 3y 34
Let x the first number. Let y the second
number.
-y -2 y 2
x y 16 (1)
2x 3y 34 (2)
2x 3y 34 2x 3(2) 34 2x
28 x 14
Check x y 16 14 2 16 16 16
Therefore, the numbers are 14 and 2.
1.2
3
Problem 2 Mike has 20 coins, in dimes and
quarters. If the total value is 2.75, how many
coins of each type are there?
10d 10q 200 10d 25q 275
Let d of dimes. Let q of quarters.
-15q -75 q 5
d q 20 (1)
10d 25q 275 (2)
10d 25q 275 10d 25(5) 275
d 15
Check d q 20 15 5 20 20 20
Therefore, Mike has 15 dimes and 5 quarters.
1.3
4
Problem 3 Sue invested 2000, part at 10 and
part at 12. If the total interest earned was
216, how much did she invest at each rate?
10x 10y 20 000 10x 12y 21 600
Let x amount at 10. Let y amount at 12.
-2y -1 600 y 800
x y 2000 (1)
10x 12y 21600 (2)
10x 12y 21 600 10x 12(800) 21 600
10x 12 000
x 1 200
Check x y 2000 1200 800 2000
2000 2000
Sue invested 1200 at 10 and 800 at 12.
1.4
5
Problem 4 It is a 230 km trip to the Jacksons
cabin. Part of the trip is on gravel roads,
where they travel at a rate of 50 km/h, and part
is on paved roads, where they travel at 80 km/h.
If the total trip is 4 h, how much time is spent
on gravel roads?
Let x time on gravel. Let y time on paved.
50x 50y 200 50x 80y 230
-30y -30 y 1
x y 4 (1)
50x 80y 230 (2)
50x 80y 230 50x 80(1) 230
50x 150 x 3
Check x y 4 3 1 4 4 4
They travelled 3 h on gravel roads.
1.5
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Problem 5 A butcher has supplies of lean beef
with 15 fat and fat trim that is 100 fat. How
many kilograms of lean beef and fat trim does
she need to make 50 kg of hamburger, which is
25 fat?
Let x of kg of beef. Let y of kg of trim.
15x 15y 750 15x 100y 1250
-85y -500 y 5.89
x y 50 (1)
15x 100y (25) x 50 (2)
15x 100y 1250 15x 100(5.89) 1250
x 44.11
Check x y 50 44.11 5.89 50
50 50
Therefore, she needs 44.11 kg of beef and 5.89
kg of trim.
1.6
7
Problem 6 A fishing boat put out to sea in the
morning travelling with the tide. It took 20 min
to cover the 6 km to the captains favourite
fishing grounds. The return trip was against the
tide and took 36 min. What was the speed of the
boat in still water and the speed of the tide?
Let x speed of the boat. Let y speed of the
current.
3x - 3y 30
x y 18
x y speed with the current x - y speed
against the current
x y 18 x - y 10
x - y 10 14 - y 10 y 4
Recall d st
2x 28 x 14
(1)
The speed of the boat in still water 14 km/h The
speed of the tide 4 km/h
(2)
1.7
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Assignment
Suggested Questions Section 1.3 Pages 26 and
27 31, 33, 34, 36, 37, 38, 43, 46
Section 1.5 Pages 39 and 40 17, 20, 24, 25, 29,
31, 32, 44-51, 56
1.8