new assignment 6 - systems of equationskillornmath.weebly.com/.../systems_of_equations_notes.pdf ·...
TRANSCRIPT
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Assignment 6 - Systems of Equations
Mr. Evan KillornKinkora Regional High School
October 19, 2014
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Table of Contents
Number of Solutions to a System of Equations
Solving Systems Graphically
Solving Systems AlgebraicallyUsing SubstitutionUsing Elimination
Appendix
Bibliography
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
A solution for a system of equations is an ordered pair (x , y) thatmust satisfy both equations.
The Left Side = Right Side, for both equations.
They are A.K.A. the point(s) of interesection on a graph.
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Number of Solutions
Both systems of Linear-Quadratic or Quadratic-Quadraticequations could have 0, 1, or 2 solutions.
Unlike when we solve radical or rational equations, there will be noextraneous roots or non-permissable values.
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Number of SolutionsNo Solutions
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Number of SolutionsOne Solution
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Number of SolutionsTwo Solutions
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Linear-Quadratic Equations Solved Graphically
In order to graph functions to find their solution(s), keep in mindtechniques that you have learned throughout this course and inprevious years.
The simplest way to graph a linear function is by using the slopey-intercept form.
y = ax + b, where a = slope, b = y-intercept
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Linear-Quadratic Equations Solved Graphically
The simplest way to graph a quadratic function is by using thevertex-form method.
y = a(x − p)2 + q, where a = stretch factor, (p, q) = vertex
If our function is written in Standard-Form, we must COMPLETETHE SQUARE to convert it to Vertex-Form.
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Linear-Quadratic Equations Solved GraphicallyExample 1
Solve the following system of equations graphically:(1) y = 2x − 4(2) y = x2 − 2x − 1
We don’t have any work to do for the linear function (1), but weneed to complete the square for the quadratic function (2).
y = x2 − 2x − 1
y + 1 = (x2 − 2x + 1) − 1
y + 1 = (x − 1)2 − 1
y = (x − 1)2 − 2
Now let’s graph the 2 functions!Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Linear-Quadratic Equations Solved GraphicallyExample 1 - Solution
Therefore, since thefunctions intersect twice,solutions to the system are:
(1,−2) and (3, 2)
If you wish to verify asolution, insert values intothe original functions.
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Quadratic-Quadratic Equations Solved GraphicallyExample 2
Solve the following system of equations graphically:(1) y = 1
2x2 + 4x + 9
(2) y = 12x
2 + 2x + 5
(1) y =1
2x2 + 4x + 9
y =1
2(x2 + 8x) + 9
y + 8 =1
2(x2 + 8x + 16) + 9
(1) y =1
2(x + 4)2 + 1
(2) y =1
2x2 + 2x + 5
y =1
2(x2 + 4x) + 5
y + 2 =1
2(x2 + 4x + 4) + 5
(2) y =1
2(x + 2)2 + 3
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Quadratic-Quadratic Equations Solved GraphicallyExample 2 - Solution
In this case, the functionsonly intersect once,therefore the solution is:
(−2, 3)
If you wish to verify thesolution, insert values intothe original functions.
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Linear-Quadratic Equations Solved Algebraically
To solve systems of equations, we must use one of two methods:
(1) Substitution or (2) Elimination
Choose whichever method is more simple to use based on theproblem.
Both of these methods simplifies two equations (with twovariables), down to one equation (with one variable ”x”).
Now solve for x, using the required method (Order of Operations,Factoring, or Quadratic Formula).
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Using SubstitutionUsing Elimination
Linear-Quadratic Equations Solved AlgebraicallyExample 1 - Substitution
Solve the following system of equations algebriacally:(1) y = 6x + 12(2) y = x2 + 2x − 9
Substitute y from equation (1)into equation (2), thensimplify:
x2 + 2x − 9 = 6x + 12
x2 + 2x − 6x − 9 − 12 = 0
x2 − 4x − 21 = 0
Solve the quadratic equationby factoring:
x2 − 4x − 21 = 0
(x − 7)(x + 3) = 0
x = 7 or x = −3
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Using SubstitutionUsing Elimination
Linear-Quadratic Equations Solved AlgebraicallyExample 1 - Substitution Solution
Now substitute your x-values, x = 7 and x = −3, back into one ofthe original equations to find your y-value.
It it always more simple to use a linear equation if possible.
(1) y = 6x + 12
y = 6(7) + 12
y = 54
(1) y = 6x + 12
y = 6(−3) + 12
y = −6
Therefore our 2 solutions are (7, 54) and (−3,−6).
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Using SubstitutionUsing Elimination
Quadratic-Quadratic Equations Solved AlgebraicallyExample 2 - Elimination
Solve the following system of equations algebraically:(1) x2 − 6x − y = −14(2) x2 − 2x − y = 10
Subtract (2) from (1) to eliminate y:
x2 − 6x − y = −14
−(x2 − 2x − y︸ ︷︷ ︸ = 10︸︷︷︸)
−4x = −24
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Using SubstitutionUsing Elimination
Quadratic-Quadratic Equations Solved AlgebraicallyExample 2 - Elimination Solution
Now solve the linear equation:
−4x = −24
−4x
−4=
−24
−4
x = 6
Substitute x = 6 into one ofthe original equations to find y:
x2 − 2x − y = 10
62 − 2(6) − y = 10
36 − 12 − y = 10
−y = −14
y = 14Therefore, the solution is (6, 14).
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Using SubstitutionUsing Elimination
It is your choice which method to use when solve any system ofequations.
It is my suggestion that you become familiar with both, but pick amethod you prefer.
For both examples, we were able to factor or use order ofoperations to solve for x. But if neither of these is possible, andthe equation is quadratic, remember that the quadratic formula isnecessary.
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Appendix
Main File - LaTeX. Second File - Maple
Grading Scheme - LaTeX = 40 marks, Maple = 10 marks
Go Beyond - I decided to use LaTex Extra content, specificallycreating a presentation. I decided to use the beamer documentclass to create this presentation in order to organize theinformation in a different format. I also found how to creategraphs in Maple with multiple functions.
Description - Due to a large amount of curriculum, my Grade 11Pre-Calculus students will complete one unit of independent study,where this presentation is their notes. The Unit is on Systems ofEquations (linear and quadratic).
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations
Number of Solutions to a System of EquationsSolving Systems Graphically
Solving Systems AlgebraicallyAppendix
Bibliography
Bibliography
Bernard, J. Blaine . Prince Edward Island Math 521B CurriculumGuide. Summerside, PE: Prince Edward Island Department ofEducation, 2012. Print.
McAskill, Bruce. ”Chapter 8 - Solving Systems of Equations.”Pre-Calculus 11. Whitby, ON: McGraw-Hill Ryerson, 2011.422-450. Print.
Mertz, Andrew, and William Slough. ”Beamer by Example.” ThePracTeX Journal 4 (2005):http://www.tug.org/pracjourn/2005-4/mertz/mertz.pdf
Mr. Evan Killorn Kinkora Regional High School Assignment 6 - Systems of Equations