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LINEAR SYSTEMS
Teacher's Guide
Linear Systems
Teacher's Guide
This teacher's guide is designed for use with Linear Systems, a series of six ten-minute
programs. The programs are broadcast by TVOntario, the television service of the Ontario
Educational Communications Authority. For broadcast dates, consult the TVOntario School
Broadcasts Calendar, which is published in September. The series is avilable on videotape to
educational institutions and nonprofit organizations.
The Guide
Writer: Ron Carr
Editor: Mei-Lin Cheung
The Series
Producer/Director: David Chamberlain
Project Officer:
Peter Puusa
Ordering Information
To order this publication or videotapes of the series, or for additional information,
please contact the following:
United States
Ontario
TVOntario
U.S. Sales Office
901 Kildaire Farm Road
Building A
Cary, North Carolina
27511
Phone: 800-331-9566
Fax: 919-380-0961
E-mail: [email protected]
TVOntario Sales and Licensing
Box 200, Station Q
Toronto, Ontario
M4T 2T1
(416) 484-2613
1.
2.
3.
4.
5.
6.
Program
The Equalizer
The Intercepts
Solving Graphically
Methods of Elimination
Substitution and Comparison
The Third Dimension
e Copyright 1994 by the Ontario Educational Communications Authority.
All rights reserved. Printed in Canada. Y160
BPN
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Page
Introduction
1
Program 1. The Equalizer
3
Program 2. The Intercepts
5
Program 3. Salving Graphically
7
Program 4. Methods of Elimination
11
Program 5. Substitution and Comparison
13
Program G. The Third Dimension
17
I NTRODUCTION
Together with numbers, equations belong to the first mathematical
achievements of civilization. Equations occur in the cuneiform texts of the
old Babylonians of about 3000 B.C. and in Egyptian papyri writings from
about 1800 B.C. In Babylon, questions and problems regarding the fair
sharing of inheritances were of utmost importance, and solutions often
involved the creation of "linear equations."
Before an algebraic symbolic language had been developed, equations had
to be written in words. The equality sign (=) in common use today, was
proposed by Robert Recorde, (1510-1588), but it took considerable time
before it was generally accepted.
The six-program series Linear Systems introduces a variety of methods for
solving systems of two equations with two variables. The graphing approach
involves finding the point of intersection of two lines, while the algebraic
methods involve elimination by addition or subtraction, substitution, or
comparison. Applications are also presented.
Linear Systems is another series in the Concepts in Mathematics group. As
in the other series, state-of-the-art computer animation techniques are used
to create interesting and informative programs.
PROGRAM I
THE EQUALIZER
Program Synopsis
The program opens in a mathematics classroom as the teacher closes the door on the way out.
Mathematical problems and solutions cover the "board." Two chairs slowly "morph" into two alien
beings, Cy and Lenny, who describe their mission to discover what Earth humans have learned.
They emphasize that "balance is the key" in much work involving mathematics.
A brief history of the need for numbers and equations is presented. A very old problem is discussed
and the resulting equation,
is written.
The concept of a linear equation with one variable is discussed and the balancing method used to
solve such equations is explained. Equivalent equations are used in this process.
Linear equations with two variables are discussed. An equation with two variables defines a relation
or relationship between the variables, and each pair of numbers is a solution. These equations have
an infinite number of solutions which are usually written as ordered pairs and are recorded in tables
of values.
At the end of the program, Cy and Lenny discuss the ability to graph the ordered pairs on a Cartesian
plane, resulting in a straight line.
Objectives
Before-Viewing Activities
After viewing this program and completing
several of the following activities and exercises,
students should be able to
1. Before viewing this program, students should
review the meaning of the following: term,
variable, constant, expression, power,
exponent, equation
• recognize an equation as being a mathematical
statement relating two equalities;
• solve a linear equation with one variable b)
isolating the variable using operations on both
sides of the equation;
• understand the concept of equivalent
equations;
• verify the solution of a linear equation with
one variable;
• understand that an equation with two variables
has an infinite number of solutions, which can
be written as ordered pairs.
2. Simplify the following expressions:
(a) 2(x + 5) - 2x - 4
(b) (x+4)(x-3)-x2
(c) (x + 2) 2 + (2x + 1)(x - 7)
3. Solve the following equations.
(a)2x+3=8
(b)3b-1=2
4. Repeat (4) for the following equations.
(a)4x+3=2(2x+1)+5
(b)5(2y+7)-3(4y-3)+2(y-22)=2
(c) 2(v + 2) = 4
(d) 3x+5 =x+4
(e) 5m-3 =7m+ 11
4. Verify the solution for each equation in (3) by
determining if the left side is equal to the right
side, by substitution.
After-Viewing Activities
5. Make up two equations, each with one
variable, such that one equation has no
solution, and the other equation has an infinite
number of solutions.
6. Solve and verify the following equation. Draw
a number line and locate the solution on the
line.
(2x+1)2 -4(x+3) 2 =2-x
1. Simplify.
(a) (2x + 3) 2 + (2x - 6)(3x + 2) - (3 - 3x)2
(b) 2x-3 _ 4x+1 - 2x+1
3
6
2
c) 2a2 - 3a2 -2a + 5a
3
4
2
2. Solve the following linear equations.
3. How many solutions does a linear equation
with one variable have? Solve the following
equations. How many solutions (roots) are
there for each equation? Express the solutions
"set-builder" form.
(a)2(x+3)+3(x+5)=5(x+1)+16
(b) 3(2 - 3a) = --2(4a + 1) - a + 8
7. Solve the equation
8. Given the equation 3x + 2y = 11, let x have
the value of 3 and solve for y.
Repeat, with x = 2,4,-3, and 0. List the related
values in a chart (table of values), and express
the solutions as ordered pairs.
PROGRAM 2
THE INTERCEPTS
Program Synopsis
Cy and Lenny continue their discussion of mathematical concepts and agree that it is too early for
them to look at systems of equations. One of the characters "morphs" into a television screen and
we are presented with a review of concepts discussed in the previous program.
A linear equation with one variable is solved by using the accepted methods of adding and subtracting
from both sides, and multiplying and dividing both sides by appropriate numbers, until the variable
is isolated on one side of the equation and the solution is on the other side.
We are reminded that a linear equation with two variables will have an infinite number of solutions,
usually represented as ordered pairs. These solutions can be tabulated using a table of values. An
equation with two variables is presented, and solutions are listed in a table of values.
The Cartesian plane is introduced with appropriate terminology (x and y axes, origin, scales). The
location of points (ordered pairs) on this plane is explained. The points in the table of values are
plotted and joined to form a (straight) line. Using the line, the concepts of the x- and Y- intercepts are
presented. Another relation is chosen and graphed using the intercepts.
Returning once again to the problem of finding ordered pairs common to two equations with two
variables, Lenny and Cy discuss the possibility of creating tables of values for each equation. These
could then be examined to try to find a match, an ordered pair common to both lines.
Objectives
Before-Viewing Activities
After viewing this program and completing
several suggested exercises; students should be
able to
1. Rearrange the following equations into the
form Ax +By+C=0.
list solutions for one equation with two
variables in a table of values;
s
construct a Cartesian plane with appropriate
labels and scales;
locate and name positions on the plane using
ordered pairs;
sketch the graph of a relation defined by a
linear equation with two variables;
find the x- and y- intercepts of a line, and use
them to sketch its graph.
(a) 4x - 5y = 6
(b) 3y + 5(2x- 3) = 0
(c)4(x-3)+3(2-2y)=x+3y-4
2. Rearrange the following equations into the
form y=mx+b.
(a) x+2y-3=0
(b) 2y-5=3x
(c) 3(1 + y) - 4(y + 1) = 9
3. List five ordered pairs for each of the
following equations:
(a)y=2x+3
(b) 2x + 3y = 6
(c) 3x - y - 9 = 0
4. What are the slopes of the lines in (3)?
5. Find the x- and y-intercepts of the lines defined
by the following equations.
(a)x+2y=2
(b) 2x - 3y = 6
4. Which form of the equation appears to be the
most appropriate to find ordered pairs?
5. Construct a Cartesian plane and locate the
following points.
A(3, 1), B(-2, 5), C(4, 0), D(3, -4),
E(-1, -6), F(1.5, 4.5)
6. Plot the ordered pairs which you found in (3a)
on a Cartesian plane. Join the points. What is
the shape of the figure?
(c)y=4x-8
(d)3y+5x-2=0
6. On one set of axes, sketch the graphs of the
lines defined by the following.
(i) x + 2y = 5
(ii) 2x + y = 4
7. Repeat (6) for the following equations.
(i) 3x + y = -3
After-Viewing Activities
1. Construct a table of values containing at least
five ordered pairs for each of the following.
(a)y=2x+3
(b) 2x + 3y = 12
2. Sketch the graphs of the relations defined by
the following equations.
(a)y=3x-1
(b) 3x - y = 3
(c) 3s - 4t= 6
3. Find three ordered pairs for each of the
following, and sketch their graphs on one
Cartesian plane.
(a)x=3
(b) y = -2
(ii) 2x - y = -10
8. (a) What characteristics of the defining equation of a relation create a graph which is
a (straight) line?
(b) Sketch the graphs of the relations defined
by the following.
(i) y = x2
(ii) y = -2x2
+
x
PROGRAM 3
SOLVING GRAPHICALLY
Program Description
Our two friends, Cy and Lenny, are still looking for an ordered pair which is common to two
equations. It has taken three days so far, and the end is still not in sight. They both realize that they
need help.
Methods used to sketch the graph of a linear relation with two variables from the previous program
are reviewed. Two equations with two variables are introduced as a system of simultaneous equations.
In attempting to solve a system such as this, we are trying to find one value for x and one value for
y, which will make each equation true, at the same time. We are looking for an ordered pair which
is common to both equations.
This is potentially very difficult and, depending on the ordered pair, perhaps almost impossible to
do by listing ordered pairs for each equation and attempting to discover a match visually. Instead,
we can see that the ordered pair common to both lines will be at the point of intersection (where the
lines cross). A second system of equations is presented and the solution is found by graphing each
line, using intercepts, and determining the coordinates of the point of intersection.
Two special cases are presented. If the two lines are parallel (and distinct), then there is no point of
intersection, and the system has no solution. Such a system is called inconsistent. If the two equations
result in two lines in exactly the same position (one line), then we have a dependent system with an
infinite number of solutions.
A final example is presented to illustrate the need for a better method to find the solution for a
system of equations. The lines intersect at a point which is not easily identified accurately.
Objectives
Before-Viewing Activities
After viewing this program and completing
several exercises and questions, students will be
able to
• understand that the method of attempting to
find a matching ordered pair from two lists of
points on two lines is not desirable;
• understand tnat the point of intersection of two
lines is the ordered pair or point which is on
both lines, and is therefore the solution for the
two equations;
• find the solution of two equations with two
variables by sketching their graphs and finding
the point of intersection by inspection;
• realize that, if two lines cross at a point which
is not easily or accurately identified, another
method to solve the equations must be found.
1. Using the intercepts, sketch the graphs of the
lines defined by the following equations.
(a)4x+y=8
(b) 3x - 2y = 6
(c)y=5x+5
2. Create tables of values for the following pair
of equations, and attempt to find the ordered
pair which is common to both tables.
(i)x+4y= 1
(ii) 2x + 5y = -1
3. Sketch the graphs of the relations defined by
the following pair of equations on one set of
axes.
(i) 4x + y = 8
(ii) 3x - 2y = -5
4. Repeat (3) for the following pair of lines.
(i) 3x - 2y =10
(ii) y = 2x - 6
5. Four times a number added to twice another
number gives a sum of 14.
(a) What are the numbers? (Is there just one
answer?)
(b) Is there a solution in which both numbers
are negative?
(c) If the first number is -5.5, what is the second number?
(d) If the second number is 4, what is the first
number?
(b) What is the y-intercept of this line?
(c) What is its slope?
(d) What is the equation of the line, in slope/
y-intercept form? (Solve for y in terms
of x.)
(e) What is true about lines which are parallel?
(f) Write the equation of the line with slope
5 and with y-intercept 3.
(g) Write the equation of a line parallel to the
line in (2a). How many such lines are
there?
(h) Sketch the graph of your line in (2g) on
the same set of axes as (2a).
(i) What is the solution of these two equations?
3. Estimate the solutions of the following
systems of equations.
(a)x+2y=3
2x-3y=1
(b)2x+5y=2
I
3x+y = 1
After-Viewing Activities
1. By sketching the graphs of the following pairs
of lines, find the solution of each system.
(a)4x+y=9
4. Classify the following systems of equations
as being consistent, inconsistent, or dependent.
(a)y=2x-3
y =3x+2
-3x + 2y = -4
(b)y=3x+6
2x-3y=-4
(b)2x+3y=6
4x+6y=4
(c) 3x + 4y = -9
x-3y=10
(c)x-3y=5
x-4y=5
2. (a) Sketch the graph of the line defined by
4x +2y=-3.
(d)y=3x+4
3x-y+4=0
(e) 3x = 2y - 2
2y=2-3x
5. Write equations which represent the following
situations. Represent the variables (unknowns)
by x and y.
(a) The sum of two numbers is 24.
(b) The difference of one number and twice
another number is 10.
(c) Mary is three years older than Bob.
(d) Five years ago, Robert was twice as old
as Razwan.
(e) The distance to Toronto is 50 km more
than twice the distance to Montreal.
(f) One airplane is travelling at a speed
300 km/h faster than another plane.
(g) Three times a number is 34 less than twice another number.
6. Solve the following equations, by eliminating
the fractions as a first step.
PROGRAM 4
METHODS OF ELIMINATION
Program Synopsis
In a continuing attempt to use the graphing method to solve a system of linear equations with two
variables, Cy and Lenny have created a coordinate system which includes numbers from -1000 to
1000, with 100 lines between each number. The grid covers the walls and ceiling of a massive
room.
The program continues with a review of previously learned material. The attempt to find an ordered
pair which is common to two linear relations would be a long search if one were to examine ordered
pairs trying to find a match. In fact, depending on the solution, the match may never be found. A
second method involving finding the point of intersection of the linear graphs of each relation is a
reasonable method, but only if the lines intersect at a point which is easily identified.
Because of these problems, an algebraic approach is discussed. (An algebraic method will produce
the accurate roots, no matter how difficult it might be to find them graphically.) The method is
called elimination (by addition or subtraction). Two equations are presented, and it is found that, by
adding the two equations, one of the variables is eliminated. The result is an equation with just one
variable which is easily solved. This value is then substituted into one of the original equations to
find the value of the other variable. It is explained that the object of any algebraic method is to
produce one equation with one variable.
A second example is produced in which addition or subtraction does not help us directly. Equivalent
equations are created by multiplying by appropriate constants. These equations can then be added
(or subtracted, if that works) to get a single equation with one variable.
A third complicated example is presented and solved. The original two equations involve fractional
numerical coefficients which can be eliminated by multiplying each equation by a common
denominator. The result is two equations which are solved as in example two.
Cy and Lenny realize that an extremely large Cartesian plane is not the answer for solving a system
of equations, and Lenny is quite upset that he must repaint the room.
Objectives
After viewing this program, and completing
several suggested questions and exercises,
students will be able to
• realize that an algebraic method of solving two
equations with two variables will produce the
roots of a system of equations, no matter how
complicated these roots may be;
• realize that the algebraic method of
elimination by addition or subtraction is
directed toward producing one equation with
one variable, which can be easily solved;
• use this method of elimination to solve
systems of simultaneous equations with two
variables;
• solve systems of linear equations which have
fractional coefficients;
solve word problems resulting in two
equations with two variables.
Before-Viewing Activities
2. Solve algebraically.
1. Solve the following systems, using the
graphing approach.
(a)x+2y= 7
3x-2y=-3
(b)2x+y=-2
3x+4y=2
(c) 5x - 3y = 15
-2x + 3y = -6
2. Estimate the roots of the following systems,
using the graphing approach.
(a)2x+3y=4
4x-y= 1
(b) x + y = -1
x-2y-4
After-Viewing Activities
1. Solve the following systems of equations
using the algebraic method of elimination by
addition or subtraction.
(a) 3x + 2y = 13
x+2y=7
(b) 5x - 3y = 17
2x+3y=11
(c)5a+2b=8
4a - 3b = -12
(d) 2m + 7n = 2
5m+3n=-2
3. Write full solutions for the following word
problems. A full solution includes opening
statements to introduce the variables, the
translation of the words into two equations
with two variables, the solution of the system,
and final statements which answer the
problem.
(a) The sum of two real numbers is 17 and
their difference is 7. Find the numbers.
(b) Three times one number, plus twice
a second number is 58. Ten times the first
number less three times the second number is -29. Find the numbers.
(c) A bag contains a total of 20 dimes and
quarters. The total value of the coins is
$3.95. How many dimes are in the bag?
(d) Tickets to a school play cost $5.00 for
adults and $3.00 for students. A total of
249 tickets is sold for $961.00. How many
adults and how many students purchased
tickets?
(e) Mary is five years older than her brother
Bill. Eleven years ago, she was twice as
old as her brother. How old are they now?
PROGRAM 5
SUBSTITUTION AND COMPARISON
Program Synopsis
When Lenny wants to substitute Brand X checkered paint for the ceiling, Cy gets the idea that
substitution might lead to another algebraic method for solving a linear system.
After the method of elimination by addition or subtraction is reviewed, the substitution method is
introduced. One of the equations is chosen from a system of two equations. This equation is solved
for y in terms of x, and this expression for y is then substituted for y in the other equation. Once
again, we obtain a single equation with one variable which is easy to solve. This value for x is then
substituted to find the value for the other variable.
This method is reviewed in a second example. It is shown that an equation can be solved for x in
terms of y, and the expression for x substituted for x in the other equation.
A fourth method, the comparison method, is presented. In this approach, both equations are solved
for one of the variables in terms of the other variable. These two expressions, for example y, are
then equated resulting in one equation with one variable.
Finally, we look at a system of three equations with three variables. To solve this system, we are
looking for values for x, y, and z which will satisfy all three equations at the same time. The elimination
method is used to reduce the system to two equations with two variables; then to one equation with
one variable. The solution to this equation is then back-substituted to find the value of a second
variable, and then the value of the third variable.
Cy and Lenny discuss the problem of graphing an equation with three variables. They suggest that
a 3-dimensional plane would be handy (sort of like they had before they repainted the room, much
to Lenny's chagrin).
Objectives
Before-Viewing Activities
After viewing this, program and working on
suggested problems, students will be able to
1. Solve the following systems using the
elimination method.
(a) 4x - 3y = 5
• . solve an equation with two variables for one
of the variables in terms of the other;
• solve a system of simultaneous equations with
two variables using the substitution method;
• solve a system of equations using the
comparison method;
• recognize when it may be most appropriate to
choose a particular one of the three available
algebraic methods;
• use the elimination method to solve a system
of three equations with three variables.
3x + 2y - 8
(b) 5x + 2y = 14
4x+7y=-5
2. Solve the following equations for y in terms
of x.
(a) 4x + y = 3
(b)3x-2y+6=0
(c)3(x-2)+5(2y+5)=9
3. Given y = 2x - 3, solve the following equations
for x.
(a) 3x + 4y = 5
(b) 2x - y = 9
2. Solve the following using the comparison
method.
(a) x+3y=-8
x-2y=12
(b) 3x + y =10
5x-y=-10
(c)-3x-5y+4=0
(c) 2x + 3y = 5
4. (a) The sum of two numbers is 188. Three
times the smaller number is 24 less than
the larger number. Find the numbers.
(b) The sum of the digits of a 2-digit number
is 10. If the number is doubled, the result
is 28 more than the number with the digits
reversed. Find the number.
(c) A parking meter contains a total of $11.20
made up from a total of 154 nickels and
dimes. How many dimes are there?
3x-2y=14
3. Solve the following using any algebraic
method of your choice. In each question, is
one method preferable over the others? Why?
(a) 3x + y = 0
4x-7y=-25
(b) 3x + 2y = 35
5x-3y=-5
After-Viewing Activities
1. Solve the following systems using the
substitution method.
(a) 3x + y = 13
(c) x+5y=-5
x-3y=11
2x+5y=13
(b)x+4y=9
3x-2y=-1
4. Solve the following systems with three
variables.
(a) x+3y+2z=15
3x-3y-2z=-11
2x+y+z=8
(c) 3x + 5y = -9
4x-3y=-12
(b) 4a-b+2c=14
3a+2b+3c=9
5a+b-7c=27
5. (a) A person who won $6000.00 in a lottery
invests part at 8.5% per year, and the remainder at 7.5% per year. How much is
invested at each rate if the total interest
earned in one year is $480.00?
(b) How many kilograms of 40% salt solution
and 30% salt solution should be mixed in
order to obtain 200 kg of 37% salt solution?
PROGRAM 6
THE THIRD DIMENSION
Program Synopsis
The program opens as one of the alien characters looks around using 3-dimensional glasses, and
expresses how impressed he is.
A system with three equations with three variables is solved by first selecting a pair of the equations
and, using elimination by addition/subtraction, eliminating one of the variables. Another pair of
equations, from the three, is then selected and the same variable is eliminated. We are left with two
equations with two variables which can be easily solved. Once the values of these two variables are
found, they can be substituted into one of the original equations to find the value of the third
variable. The solution is then expressed as an ordered triple.
The 3-dimensional coordinate system is introduced and the locations of two ordered triples are
found. When these points are joined in 3-space, the result is a line.
It is explained, with accompanying illustrations, that two lines in 3-space may intersect at one
point, or they may be parallel with no point of intersection. A third situation is possible as well. In
this case the two lines are not parallel, and do not intersect. They are called skew lines.
The graph of a linear equation with three variables is discussed. Ordered triples are found and listed
in a table of values. When these points are graphed, the result is a flat surface or plane. The intercepts
of the plane are also discussed.
If two planes are not parallel, the intersection is a line. When a third plane is introduced, it may
tersect this line at one point. This is the solution of the three equations with three variables with
which we started. It is also shown that three planes can be situated in other situations. One result
gives a line as the solution, and another placement shows that there is no intersection and no solution
to the linear system.
The program ends as Cy and Lenny "morph" back into chairs as the class is set to resume.
Objectives
After viewing this program and completing
several suggested exercises and questions,
students will be able to
• construct a 2-dimensional representation of a
3-dimensional space;
• locate the positions of ordered triples in 3space;
• recognize that a linear equation with three
variables represents a plane in 3-space:
• find the intercepts of a plane, and use these to
draw a representation of the plane;
• recognize that the solution of a linear system
with three variables is the intersection of three
planes, and that these three planes may
intersect in a single point, a line, or not at all.
Before-Viewing Activities
1. Find the x- and y-intercepts of the following
lines.
(a) 2x + 3y = 9
(b) 3x-5y+8=0
(c) 4x = 7
(d) 3y -12 = 0
2. Solve the following systems of three equations
with three variables.
(a) 2x + y + 2z =15
(b) Locate the points A(1, 2, 4), B(2, -3, 5),
and C(0, -2,-3).
2. (a) Locate the points D(1, 2, 3) and
E(-2, -1, 4).
(b) Draw the line DE.
(c) Draw a second line which is parallel to DE.
(d) Draw a line which is not parallel to DE
and which does not intersect DE.
3. Find the x-, y-, and z-intercepts of the
following planes.
3x+y-3z=-9
(a) 2x+y+3z=6
4x+y+5z=32
(b) 3x-2y-2z=-6
(c) 4x+2y-z=4
(b) 2x + 3y + 4z = -20
3x - 2y + 2z = -1
4x + 5y + 5z = -26
3. Solve.
2x-y=6
4x-2y=8
Are these lines parallel or coincident?
4. Solve.
-3x+6y-9=0
x-2y=-3
Are these lines parallel or coincident?
After-Viewing Activities
1. (a) Construct a diagram of a 3-dimensional
space (on your 2-dimensional page).
4. On separate graphs, sketch the planes defined
in (2).
5. On a plain page, draw the following:
(a) two planes which intersect in a line
(b) two planes which are parallel
(c) three planes which are parallel
(d) three planes which intersect at a single
point
(e) three planes which intersect in a single
line