### MATHEMATICS 2204

```MATHEMATICS 2204
Unit 01
Investigating Equations in 3-Space
Solving Systems of Equations
Involving Two Variables
Choosing a phone plan
SOLVING SYSTEMS OF EQUATIONS
INVOLVING TWO VARIABLES
developing equations relating 2 variables
and the corresponding graphs - cdli - math
2204.swf
FINDING POINTS OF INTERSECTION
• finding points of intersection algebraicly - cdli
- math 2204.swf
SEATWORK
Complete Investigation 1 on pages 2 & 3 of your
text.
• Do Investigation Question 1 AND also answer the
following questions:
a) For what number of minutes of phoning would it be
best to have Plan A?
b) For what number of minutes of phoning would it be
best to have Plan B?
c) For what number of minutes of phoning would it be
best to have Plan C?
HOMEWORK
• Do CYU Questions 2 - 4 on page 3.
Visualization in Three
Dimensions
Labelling Axes in Three Dimensions
CLASSROOM AS A MODEL OF 3-SPACE
•For a point (x , y , z), the z-axis
(the dependent variable) is
always vertical, with positive in
the up direction
•the y-axis is always to the left
and right, with positive to the
right
•the x-axis is always depicted as
coming out of the page toward
you. with positive coming out
and negative "back behind the
page".
CLASSROOM AS A MODEL OF 3-SPACE
• Stripping away the "room" above gives the
following set of axes, the dashed lines are the
parts you can't see in the above diagram:
3 – SPACE COORDINATE SYSTEM
CONVENTIONS
• For a point (x , y ,z), the z-axis (the dependent
variable) is always vertical, with positive in
the up direction;
• the y-axis is always to the left and right, with
positive to the right;
• the x-axis is always depicted as coming out of
the page toward you. with positive coming out
and negative "back behind the page".
INTRODUCING 3 SPACE
• introducing three space - cdli - math 2204.swf
EQUATIONS IN 3 VARIABLES
• writing equations in three variables - cdli math 2204.swf
Visualization in Three
Dimensions
Sketching Planes on Paper
INTERACTIVE
• constructing planes in 3 space using cubes cdli - math 2204.swf
SKETCHING PLANES ON ISOMETRIC
PAPER
sketching planes on isometric paper cdli - math
2204.swf
A TWO DIMENSIONAL DRAWING OF A
THREE DIMENSIONAL OBJECT
diagrams on page 14 & 15
USING THE INTERCEPT METHOD TO
SKETCH A PLANE
• One method of graphing an equation in two
variables such as y = 2x - 6 was to use the twointercept method.
– To find the y-intercept, just substitute x = 0, in the
above equation, this gives y = 2(0) - 6 or y = -6.
• So the y-intercept is (0 , -6).
– Similarly, if we substitute y = 0 into the equation we
have 0 = 2x - 6, and rearranging this gives x = 3.
• So the x-intercept is (3 , 0).
– Plotting these points on a set of axes and drawing the
line through them gives the following graph:
USING THE INTERCEPT METHOD TO
SKETCH A PLANE
GRAPH A PLANE ON A SET OF THREE
DIMENSIONAL AXES USING THE
INTERCEPT METHOD
• Consider the equation z = -2x -3y +6,
– the z-intercept (substituting x = 0, y = 0) is 6
– the y-intercept (substituting x = 0, z = 0) is 2
– the x-intercept (substituting y = 0, z = 0) is 3
PLOTTING POINTS IN 3 SPACE
• plotting points in three space - cdli - math
2204.swf
GRAPHING A PLANE ON A SET OF
THREE DIMENSIONAL AXES USING THE
INTERCEPT METHOD
EXAMPLE
• Use the intercept method to sketch the plane
with equation 3x + 6y - 2z = 6
HOMEWORK
• Do Focus A: Investigation 3: Visualizing the
Phone Charges Page 9
• Do Focus Questions 1 & 2 on page 10.
• Complete Investigation 3 on pages 10 - 12.
• Do Investigation Questions 3 - 8 on pages 12
and 13.
• Do CYU Questions 9 -11 on page 13.
Solving systems of equations
involving two and three variables
Combining information from
different equations
SUBSTITUTION METHOD
• One method used to solve two equations with
two variables (unknowns).
SUBSTITUTION EXAMPLE
• Page 1
• Solve:
• Page 2
EXAMPLES DONE IN CLASS
• Solve the following systems using the
substitution method:
a) 2x + 5y = 1
-x + 2y = 4
b) 0.29k + 9d = 119
0.10k + 29d = 300
INTERACTIVE
• focus d - math 2204.swf
HOMEWORK
• Do the CYU Questions 5 to 11 on pages 26 and
27.
Solving systems of equations
involving two and three variables
Graphing equivalent systems of
equations
CREATING AND ANALYZING
EQUIVALENT SYSTEMS OF EQUATIONS
Rearrange each equation to get its slope and y-intercept
and sketch the graph of the system.
CREATING AND ANALYZING
EQUIVALENT SYSTEMS OF EQUATIONS
REARRANGE EACH EQUATION TO GET ITS SLOPE AND YINTERCEPT AND SKETCH THE GRAPH OF THE SYSTEM.
CREATING AND ANALYZING
EQUIVALENT SYSTEMS OF EQUATIONS
THE GRAPH OF THE SYSTEM
ELIMINATION
• Another algebraic method for solving two
equations with two variables (that is, finding
their intersection point) is the Elimination
Method.
ELIMINATION
We could either eliminate y by multiplying the first equation by
3 and add or we could multiply the second equation by -2,
which gives -2x - 6y = -18, and add. Let's choose the second
option:
GRAPHING THE RESULTING EQUATION
ON THE SAME AXES AS THE ORIGINAL
SYSTEM
EXAMPLE DONE IN CLASS
• Graph the following system of equations:
• Draw a vertical line and a horizontal line
through the point of intersection of the two
lines.
EXAMPLE DONE IN CLASS
• Solve by the method of Elimination:
EXAMPLE DONE IN CLASS
Solve by the method of Elimination:
SEATWORK AND HOMEWORK
• Complete Investigation 4 on pages 27 & 28 of
• Do Investigation Questions 12-14 on page 28.
Pay particular attention to Question 13, it
shows the main point of the lesson.
• Do the CYU Questions 15 - 21 on pages 28 and
29.
SEATWORK AND HOMEWORK
• Complete Investigation 5 on page 30 of your
text.
• Do Investigation Questions 22 - 27, on page
31.
• Do the CYU Questions 28 - 31 on pages 31 32.
SEATWORK AND HOMEWORK
• Complete Investigation 6 on pages 32 & 33 of
• Do Investigation Questions 33 -35 page 33.
• Do the CYU Questions 36 - 45 pages 34 - 35.
Solving Systems of Equations
Using Matrices:
Writing Equations in Another Form
MATRIX MULTIPLICATION EXAMPLE
• Notebooks cost \$1.19 and a pen costs \$1.69
– Susan bought 5 notebooks and 3 pens
– Mary bought 4 notebooks and 2 pens
– Art bought 3 notebooks and no pens
How much each person spent can be represented by the
following matrix multiplication:
MATRIX MULTIPLICATION EXAMPLE
• Recall that to multiply two matrices, the
number of columns in the left matrix must
equal the number of rows in the right matrix.
• In our example, the left matrix has 2 columns
(it is a 3 x 2 matrix) and the right matrix has 2
rows (it is a 2 x 1 matrix).
MATRIX MULTIPLICATION EXAMPLE
• The product matrix is shown below:
WRITING SYSTEMS IN MATRIX FORM
• write one matrix containing the coefficients of the
variables, one containing the variables, and one
containing the constant terms.
ANOTHER MATRIX MULTIPLICATION
EXAMPLE
• To write the matrix form of a system of equations, there must be
the same number of rows as there are variables.
• If any variables or equations are missing, they must be filled in with
zero coefficients.
EXAMPLE DONE IN CLASS
• Write the following systems in matrix form:
EXAMPLE DONE IN CLASS
• Write the following systems in matrix form:
HOMEWORK
• Do the CYU questions 5 - 6 on page 38.
Solving Systems of Equations Using
Matrices:
Solving a Matrix Equation
SOLVING REGULAR EQUATIONS
In that case, we multiplied both sides of the
equation by the inverse of the coefficient which
gave:
Notice that we wanted to get the coefficient of
x to be 1, the identity for multiplication.
SOLVING MATRIX EQUATIONS
First write the equation in matrix form:
We now multiply both sides by the inverse of
the coefficient matrix
SOLVING MATRIX EQUATIONS
the equation can be written as:
Since the two matrices on the left are inverses of
each other, their product will give the identity
matrix and our equation will look like this:
SOLVING MATRIX EQUATIONS
Use ti-83 to get inverse of matrix
THE INVERSE MATRIX
SOLVING MATRIX EQUATIONS
SOLVING OUR ORIGINAL EQUATION
GIVES:
EXAMPLE DONE IN CLASS
• Use your TI-83 calculator to find the inverse of the
following matrix:
EXAMPLE DONE IN CLASS
• Use your TI-83 calculator to find the inverse of
the following matrix:
EXAMPLE DONE IN CLASS
• Solve the following system using matrices:
HOMEWORK
• Complete Investigation 7 on pages 38 & 39 in
• Do the Investigation Questions 7 to 10 on
page 40.
Solving Systems of Equations Using
Matrices:
Predicting the Inverse of a 2 x 2
Matrix
INTERACTIVE
• finding inverses using discriminants - math
2204.swf
TO FIND THE INVERSE OF A 2 X 2
MATRIX
• you can switch the numbers on the major
diagonal, negate the other two numbers, then
divide all values by the determinant.
EXAMPLE DONE IN CLASS
• Find the determinant for the following matrix:
EXAMPLE DONE IN CLASS
• Calculate the inverse of the following matrices then
EXAMPLE DONE IN CLASS
• Calculate the inverse of the following matrices then use
Using Equations for Predicting:
Applications of Systems and Matrices
INTERACTIVE
• defining equations for parabolas - math
2204.swf
EXAMPLE DONE IN CLASS
• Determine the equation of the parabola that
passes through the following three points:
(-2 , -5), (0 , -3), and (2 , 3).
HOMEWORK
• Do CYU Questions 3 to 7 on pages 48 & 49.
REVIEW
• Study the Summary of Key Concepts on pages
54 to 65.
• Do Practice Exercises 1 to 20 on pages 66 to
67.
END OF UNIT 1
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