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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



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Table of Contents

Number of Solutions to a System of Equations

Solving Systems Graphically

Solving Systems AlgebraicallyUsing SubstitutionUsing Elimination

Appendix

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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.




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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.




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Number of SolutionsNo Solutions




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Number of SolutionsOne Solution




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Number of SolutionsTwo Solutions




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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




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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.




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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



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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.




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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




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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.




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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).




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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




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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).




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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




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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).




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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.




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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).




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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


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