Expectations: i. solve systems of equations using substitution and elimination.

For more activities and resources from the University of Waterloos Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 1

ANALYTIC GEOMETRY : SOLVING LINEAR SYSTEMS OF EQUATIONS This resource may be copied in its entirety, but is not to be used for commercial purposes without permission from the Centre for Education in

Mathematics and Computing, University of Waterloo.

Play the Rags to Riches http://www.quia.com/rr/59310.html first.

You may also go to www.wiredmath.ca for the link.

1. Solve each system of equations by substitution.

a.

b. c.

d.

e. f.

g.

h. i.

2. Solve each system of equations by elimination.

a.

b. c.

d.

e. f.

3. A barn dance fundraiser raises $2,050 for a school. A total of 125 tickets were

sold. The tickets cost $20 for adults and $14 for students. The relationship

between the number of adult tickets sold, a, and the number of student

tickets sold, s, can be modeled by the following system of equations.

125

20 14 2050

a s

a s

Find the number of adult tickets and student tickets sold.

4. Two numbers have a difference of 4. The sum of triple the larger number and four times the smaller

number is 54. What are the numbers?

2 9

2 1

y x

y x

3 2

8 2 2

x y

x y

2 1

9 4 2

y x

x y

3 3 11

5

x y

x y

3 5 14 0

3 2 0

x y

x y

4 8

7 2 1

a b

a b

7 3

14 6 2

x y

x y

1

6 3

32 4

x y

x y

3 7 0

15 5 13 0

a b

a b

2 3 6

2 3 6

x y

x y

7 2 3

4 4 1

x y

x y

3 6 7 0

9 18 9 0

y x

y x

0.4 0.3 0.1

0.2 0.5 0.3

x y

x y

3 9

4 36 12

x y

y x

3 11

4 3 2

2 3 21

5 2 10

yx

x y

Expectations: i. solve systems of equations using substitution and elimination.

For more activities and resources from the University of Waterloos Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 2

5. Jason and Greg go to a caf with their friends. If they order two small hot

chocolates and two cookies, their bill totals $4.40. If they order three small hot

chocolates and four cookies, their bill totals $7.50. How much does a small hot

chocolate cost? How much does a cookie cost?

6. Three lines 2 1 0y x , 3 7 10 0y x , and 2 17 50 0y x intersect to form a triangle. Find the

coordinates of the vertices of the triangle.

7. Solve the following system of equations. Geometrically, what does the result mean?

8. Solve the system of equations.

9. The solution to each linear system is ( 1, 2). Determine the values of m and n.

a.

b. c.

CHALLENGE YOURSELF!

10. Solve this system of equations for x and y by letting 1

ax

and 1

,by

0, 0.x y

A Slice of History

Gauss developed a method of elimination that used matrices

to solve least squares problems relating to space and astronomy.

He later used it while making measurements of the earth and

its surface.

3 7

3 4 5

x my

nx y

9

7 2 3

mx ny

x ny

6 3 0

2 5 8

ny mx

ny mx

3 2 2

3

8 4 2

9

x y

x y

3 13 0

4 2 4 0

5 3 27 0

x y

x y

x y

2 1 3 2(1) 2

5 4

3 1 5 3(2) 14

5 3

x y

x y

Expectations: i. solve systems of equations using substitution and elimination.

For more activities and resources from the University of Waterloos Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 3

11. Chuck ran 8 km on the treadmill at the gym. He ran at an average speed of 9 km/h

and walked at an average speed of 5 km/h for the rest. He spent 30 minutes more time

running than walking. How long did it take him to finish his workout?

12. Solve each system of equations of three or four variables.

a.

b.

c.

d. e.

f.

EXTENSIONS

13. Nancy is travelling with her family to Florida for vacation. To fly, there are weight

restrictions on her baggage. If she brings two identical carry-on bags and one

checked baggage, the total weight is 69.5 lbs. If she brings one carry-on and two

identical checked baggage, the total weight is 100 lbs. Nancys bags are full. The airline has a maximum weight of 50 lb for checked bags and 22 lb for carry-on

baggage.

a. Determine the weights of Nancys carry-on bag and checked baggage. b. Does Nancys carry-on and checked baggage meet the guidelines?

14. If the system of equations

has the solution (2, 1, 1) , find the values of a, b, and c.

15. Solve the system of equations to find x, y, and z.

(Hint: Let 2 2 2, , and .d x e y f z Find d, e, and f first.)

16. Solve for x and y.

a.

b. c.

0

2 6

3

y z

x y z

x y

1

2

3

x y

y z

z x

7

2 2

4

2 4 19

y z

y z

w x y z

w x y z

1 2

1 2

3 4

3 4

2

4

2

4

x x

x x

x x

x x

3 2 9

2 5 3

3 6 2 8

x y z

x y z

x y z

6

1

1

ax by cz

bx cy az

cx ay bz

2 2 2

2 2 2

2 2

9

3 7

2 6

x y z

x y z

y z

ax by c

ax by d

1

1

gx hy

hx gy

2 2( ) ( ) 2( )

2

p q x p q y p q

x y p

2

2 1

3

y z

y z

x y z