MATHEMATICS 2204 ST. MARY’S ALL GRADE SCHOOL. Unit 01 Investigating Equations in 3-Space.

MATHEMATICS 2204

ST. MARY’S ALL GRADE SCHOOL

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 two-intercept 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 = 1190.10k + 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.


REARRANGE EACH EQUATION TO GET ITS SLOPE AND Y-INTERCEPT AND SKETCH THE GRAPH OF THE SYSTEM.


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.


• Solve by the method of Elimination:


Solve by the method of Elimination:

SEATWORK AND HOMEWORK

• Complete Investigation 4 on pages 27 & 28 of your text.

• 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.


• 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.


• Complete Investigation 6 on pages 32 & 33 of your text.

• 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:


• 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).


• 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.


• Write the following systems in matrix form:

HOMEWORK

• Do the CYU questions 5 - 6 on page 38.


Solving a Matrix Equation

http://www.cdli.ca/courses/math2204/unit01_org04_ilo02/a_getready.html

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 EQUATIONSthe 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:


Use ti-83 to get inverse of matrix

THE INVERSE MATRIX


SOLVING OUR ORIGINAL EQUATION GIVES:

EXAMPLE DONE IN CLASS• Use your TI-83 calculator to find the inverse of the following

matrix:


• 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 your text.

• Do the Investigation Questions 7 to 10 on page 40.


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.


• Find the determinant for the following matrix:


• Calculate the inverse of the following matrices then use your TI-83 to check your answers:

Using Equations for Predicting:

Applications of Systems and Matrices



INTERACTIVE

• defining equations for parabolas - math 2204.swf


• 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

MATHEMATICS 2204 ST. MARY’S ALL GRADE SCHOOL. Unit 01 Investigating Equations in 3-Space.

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