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May 23, 2012
MPM2D1
Name: ______________________
NOTE: A great way to study is to turn to the examples given in the text and try to solve them without looking at
how it is done in the text. Once you have finished working through the problem by yourself, look through the
solution outlined in your text book!
Chapter 1 Review – Linear Systems
2
Ordered Pairs and Solutions
CONCEPTS
EXAMPLE PROBLEMS
Ordered Pair – A pair of numbers used to name a point
on a graph, such as (-5, 3).
System of Equations – Two or more equations studied
together. x  y  3 and 3 x  y  1 is a system of
Identify the ordered pair that satisfies both
equations.
(a) x  y  3
(0, 3), (1, 2), (2,1)
2x  y  0
equations.
1.1 Connect English With Mathematics And Graphing Lines
CONCEPTS
Problem solving involves translating from words into
equations. The following chart gives an indication of what
words translate into which operations:
Addition
increased by
more than
combined, together
total of
sum
added to
Subtraction
decreased by
minus, less
difference between/of
less than, fewer than
Multiplication
of
times, multiplied by
product of
increased/decreased by a
factor of (this type can
involve both addition or
subtraction and
multiplication!)
Division
per, a
out of
ratio of, quotient of
percent (divide by 100)
Equals
is, are, was, were, will be
gives, yields
sold for
from: http://www.purplemath.com/modules/translat.htm
EXAMPLE PROBLEMS
Two gears have a total of 112 teeth. One of the gears
has 8 less than twice the number of teeth on the
other gear. Find the number of teeth on each gear.
At a movie house, 3 small drinks and 2 medium drinks
cost $6. Two small drinks and four medium drinks
cost $8. Find the price of each size of drink.
At a campground, there is a charge per campsite
occupied and also a charge per person camping. For
one site with four people the total cost is $8.25. A
group of nine people are charged $17.75 for occupying
two sites. Find the charge per site and per person.
The admission fee at a small fair is $1.50 for children
and $4.00 for adults. On a certain day, 2200 people
enter the fair and $5050 is collected. How many
children and how many adults attended?
May 23, 2012
MPM2D1
Name: ______________________
To graph a linear relation, make a table of values and plot
at least 2 points. Then draw a line to join the points. The
line represents the relation.
Solve the system by graphing and check your
solution.
5 x  2 y  10
2  2y  2
The equations:
x  2 y  1  0 and
x  2 y  5  0 are shown
at right. The lines appear
to intersect at the point
(-2, 1.5). Check the
solution by filling these
values into both
equations:
x  2 y 1  0
 2  2(1.5)  1  0
x  2y  5  0
 2  2(1.5)  5  0
 2  3 1  0
 235  0
00
00
Intersecting lines have different slopes. This results in
ONE solution.
Parallel and distinct lines have the same slope and
different y-intercepts. This results in NO solutions.
Equations which graph the same line or coincident lines
have the same slope and the same intercepts. This
results in an INFINITE number of solutions.
The Phoenix Health Club charges a $200 initiation
fee, plus $15 a month. Champion Health Club
charges a $100 initiation fee, plus $20 a month.
The costs can be compared using the following
equations.
Phoenix Cost: C  200  15m
Champion Cost: C  100  20m
(a) Find the point of intersection of the two lines.
(b) After how many months are the costs the
same?
(c) If you joined a club for only a year, which club
would be less expensive?
Without graphing, determine whether the system
has one solution, no solution, or infinitely many
solutions.
2 y  3x  1
8 y  4  12 x
Put the equation into slope -- y-intercept form to
compare slopes and y-intercepts.
Key Concepts – page 16.
See examples on pages 9 to 13.
1.2 The Method of Substitution
CONCEPTS
EXAMPLE PROBLEMS
 If two lines intersect, then there will be a point in
common between them.
 Two lines will intersect if their slopes are different.
Find the solution to the linear system
 To find the point of intersection (the solution)
1.
solve one equation for one of its variables
2. substitute the solved variable’s expression into
the other equation
3. solve for the variable
4. use the value of the variable to solve for the
second variable
5. State the answer.
EXAMPLE:
Solve: 2 x  y  13
4x  3y  1
3e  f  2  0
5e  2 f  3
The arms of an angle lie on the lines y 
2
x7
3
and 3x  2 y  12 . What are the coordinates of
the vertex of the angle?
May 23, 2012
MPM2D1
1.
2.
3.
4.
Name: ______________________
2 x  y  13
 y  13  2 x
y  13  2 x
4x  3y  1
4 x  3(13  2 x)  1
4x  39  6x  1
10 x  40
x4
y  13  2 x
y  13  2(4)
y  5
The point of intersection is (4, -5).
Key Concepts – page 25.
5.
See examples on pages 21, 22, 23, & 25.
1.3 Investigate Equivalent Linear Relations & Equivalent Linear Systems
CONCEPT
 Equations can appear to be different, but once graphed, it is obvious that they will graph the same line. These
are called equivalent linear equations, or equivalent linear relations. An example of equivalent linear equations
is the line y  2 x  8 and
1
y  x4.
2
When 2 pairs of linear equations have equivalent linear systems, they will have the same point of intersection.
These sets of 2 lines will each graph the same 2 lines, so… they have equivalent linear equations/relations.
Key Concepts – page 31.
See Communicate Your Understanding on page 32.
1.4 The Method of Elimination
CONCEPTS
1.
2.
3.
4.
5.
6.
It is easiest if decimals and fractions are cleared
first (multiply left side & right side by denominator
to clear fractions).
Line up like terms in the same columns by rearranging the equations.
Either x coefficients OR y coefficients must be the
same value preferably with opposite signs.
Add or subtract matching terms in the two
equations – this is how one variable is eliminated
Solve for the variable which is left
Substitute the value into the equation to solve for
the second variable (just like we did in section 1.3)
EXAMPLE PROBLEMS
Solve and check:
5x  2 y  5
3x  4 y  23
May 23, 2012
MPM2D1
7.
State the result.
EXAMPLE:
y 11
3
x 
4
3 2
2x 3y
21


5
2
10
9 x  4 y  66 (multiplied eq. 1 by 12)
4 x  15 y  21 (multiplied eq. 2 by 10)
Solve:
1.
2.
3.
4.
5.
6.
Equations are already lined up properly.
36 x  16 y  264 (multiply eq. 1 by 4)
36 x  135 y  189 (multiply eq. 2 by 9)
151y  453
(result of subt.)
y3
3
3 11
x 
4
3 2
3
11
x  1
4
2
9
3x   4
2
18
x
3
x6
The point of intersection is (6, 3).
Key Concepts – page 39.
Name: ______________________
Solve and check:
2
1
x  y  2
3
5
1
1
x  y  7
3
2
Word Problems:
Two gears have a total of 112 teeth. One of the gears
has 8 less than twice the number of teeth on the
other gear. Find the number of teeth on each gear.
At a movie house, 3 small drinks and 2 medium drinks
cost $6. Two small drinks and four medium drinks
cost $8. Find the price of each size of drink.
At a campground, there is a charge per campsite
occupied and also a charge per person camping. For
one site with four people the total cost is $8.25. A
group of nine people are charged $17.75 for occupying
two sites. Find the charge per site and per person.
The admission fee at a small fair is $1.50 for children
and $4.00 for adults. On a certain day, 2200 people
enter the fair and $5050 is collected. How many
children and how many adults attended?
See Examples on pages 35, 36, 37, 38.
1.5 Solve Problems Using Linear Systems
CONCEPTS
EXAMPLE PROBLEMS
For each problem:
1.
Write the “let” statements for 2 different
variables (Let x represent… Let y represent…)
2. Write the 2 equations using your variables. You
may wish to organize your information in a chart
similar to the ones below.
3. Solve for your variables using methods of
substitution or elimination.
4. Write your concluding statement to answer the
question being asked.
5. Make sure that you answered what was asked and
you used the proper representation of x and y.
Investing Money / mixing solutions:
Kelsey invested in a GIC that paid 6% interest per
annum and in a Canada savings bond that paid 4%
interest per annum. If he invested a total of
$14 000 and earned $780 interest in a year, how
much did he invest at each rate?
How many grams of pure gold and how many grams
of an alloy that is 55% gold should be melted
together to produce 72 g of an alloy that is 65%
gold?
Distance, speed, time:
A plane flew 3000 km from Calgary to Montreal
with the wind in 5 hours. The return flight into
May 23, 2012
MPM2D1
Name: ______________________
Investing money / mixing solutions:
__%
__%
total / results
Money Invested ($)/
Volume of Solution
Interest Earned ($) /
Volume of Pure Solution
D
Distance 
S T
You drive 210 miles to a relative's house. It takes
you 4 hours. Part of the time you're on a freeway,
where the speed limit is 60 mph. The rest of the
time you're on smaller roads, where the speed
limit is 30 mph. Supposing you drove exactly at
the speed limit the whole way, how much time did
you spend on each type of road?
speed  time
Distance, Speed, Time:
Distance
(km)
Speed
(km/h)
the wind took 6 hours. Find the wind speed and
the speed of the plane in still air.
Time
(h)
Equations
D=ST or TOTALS
into the wind /
Part A of trip
against the wind /
Part B of trip
Area & perimeter:
If one side of a square is doubled in length and the
adjacent side is decreased by two centimetres,
the area of the resulting rectangle is 96 square
centimetres larger than that of the original
square. Find the dimensions of the rectangle.
If the height of a triangle is five inches less than
the length of its base, and if the area of the
triangle is 52 square inches, find the base and the
height.
See Examples on pages 43(numbers), 44(dist, speed, time), 45(mixtures).
Know how to find the perimeter of different shapes.
Know how to find the area of different shapes.
Key Concepts – page 46.
See also:
Review on pages 48-49.
Chapter Test on pages 50-51.
http://www.purplemath.com/modules/index.htm (see “Solving Word Problems” “distance”, “investment”,
“mixture”, “number” -- solutions are given)