linear systems - ch. 1.1, 1.2, 1.3 (part)

8/4/2019 Linear Systems - Ch. 1.1, 1.2, 1.3 (Part)

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Section One: Linear Systems

Textbook: Ch. 1.1, 1.2, 1.3 (part)

GOALS OF THIS CHAPTER

- define a linear system

- review the elimination method to solve linear systems

- review the substitution method of solving linear systems

- see examples of systems that have unique solutions, infinitely manysolutions, and no solution

- give a geometric interpretation of the solution


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INTRODUCTION

An unknown, or variable, is a letter we are trying to solve for.

Lets start with some definitions:

A linear equation is an equation that relates unknowns and

numbers. The power of all unknowns must be equal to one.


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INTRODUCTION

2x = 17

Ex. 1 Some Linear and Non-Linear Equations

3x2 + y = 7

y = 3x + 22 x= y-4 + 3

linear

linear

non-linear

non-linear


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INTRODUCTION

x + 3y - z = 17x + z = 7

4y 3z = 9

Ex. 2 Some Linear Systems

2x - 3y = 16x + y = 1

-x + 4y = 2/3

A system of 3 equations in3 unknowns.

A system of 3 equations in2 variables.


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INTRODUCTION

We can classify solutions as inconsistent or consistent:

Consistent Inconsistent

Unique Soln Yes No

Infinite Soln Yes No

No Soln No Yes


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SUBSTITUTION METHOD

This method is best used when we have a system of two equations intwo variables.

x + 2y = 102x - 2y = -4

1

2

Label your equations with a number.

Isolate a variable in equation one.x = -2y +102x - 2y = -4

1

2=

Ex. 3 The Substitution Method


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SUBSTITUTION METHOD

Substitute x = -2y + 10 into the secondequation.

x = -2y +102(-2y+10) - 2y = -4

1

2=

x = -2y +10-4y + 20 - 2y = -4

1

2= Expand into the brackets.

x = -2y +10-6y = -24

1

2= Collect like terms.


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SUBSTITUTION METHOD

x = -2(4) +10y = 4

1

2=

Divide both sides by -6.x = -2y +10

y = 4

1

2=

Substitute y = 4 into the first equation.

x = 2y = 4

1

2=

Solution set is {x = 2, y = 4}. This is a uniqueand

consistentsolution.


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ELIMINATION METHOD

This method can be used when your linear system has any number ofequations and any number of unknowns.

We can perform any of the three following steps:

(1) Interchange any two equations.(2) Multiply an equation by a non-zero number.

(3) Add (or subtract) a multiple of one equation to (from) another.


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ELIMINATION METHOD

Lets see what these steps may look like using a 2 by 2 system.

x + 2y = 102x - 2y = -4

1

2

2x - 2y = -4x + 2y = 10

1

2

Ex. 4 Elimination Steps

Switching equation one and two.

One switched with two.

1 2


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ELIMINATION METHOD

x + 2y = 102x - 2y = -4

1

2


Multiplying equation one by two.

One goes to two times one.

1 12 *1

2

2x + 4y = 202x - 2y = -4


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ELIMINATION METHOD

x + 2y = 102x - 2y = -4

1

2


Adding equation two to equation one.

One goes to one plus two.

1 1 + 21

2

3x = 62x - 2y = -4


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Lets actually solve this problem now!

x + 2y = 102x - 2y = -4

1

2

Indicate what step you are doing over anarrow.

Ex. 5 The Elimination Method1 1 + 2

1

2=

3x + 0y = 62x - 2y = -4

1 1 / 3

ELIMINATION METHOD

If we divide by three, we will get ananswer for x.


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1

2

=

Ex. 5 The Elimination Method

x = 2

2x - 2y = -4

2 2 / 2

1

2=

x = 2x - y = -2

2 2 - 1

1

2=

x = 2- y = -4

2 2(-1) * 1

2

x= 2y = 4

Our solution set is {x = 2, y = 4}.

ELIMINATION METHOD

Instead of substituting, we now try to

eliminate x in the second equation.


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ELIMINATION METHOD

There are three types of systems we will work with in this course.

An equally determined system has the same number of equations asvariables. These systems can have any type of solution.

x + 3y - z = 17x + z = 7

4y 3z = 9

A system of 3 equations in 3unknowns. It is equally determined.


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ELIMINATION METHOD

An overdetermined system has more equations than unknowns.These systems can have any type of solution; however, they are morelikely to have no solution. This is because the more equations we have,

the less leeway the variables have.

2x - 3y = 16x + y = 1

-x + 4y = 2/3

A system of 3 equations in 2variables. It is overdetermined.


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ELIMINATION METHOD

An underdetermined system has more variables than equations.These systems cannot have a unique solution. If they have a solution,it will be an infinite solution. This is because the fewer equations we

have, the more leeway the variables have.

x + 3y - z = 17x + z = 7

A system of 2 equations in 3unknowns. It is underdetermined.


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ELIMINATION METHOD

Next, we do some more examples.

Ex. 6 A 3 by 3 Unique Solution

Ex. 7 A 3 by 2 Inconsistent Solution

- done on overhead

- done on overhead

Ex. 8 A 2 by 3 Infinite Solution

- done on overhead


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ELIMINATION METHOD

Tips and Tricks:

Try to multiply or divide equation one so that the coefficient of xmatches the coefficient of x in equation two. This way, you can addor subtract the two equations to eliminate the x variable.

If you notice that two equations have the same coefficient of avariable, you can add or subtract right away to eliminate thatvariable.

Learn what type of solution to expect from what type

of system and learn how to double-check your answer!

TRY TO LEARN THE METHOD SHOWN IN CLASS(ARROWS AND LABELLING). IF YOU DO NOTLEARN THIS METHOD EARLY, YOU WILL BE IN BIG

TROUBLE IN A FEW CHAPTERS.


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GEOMETRIC INTERPRETATION

When dealing with three or two variables, we can

visualize the system as well as the solution set.First, lets look at systems with two unknowns.

Unique Solution Lines intersect at

a single point.

Infinitely ManySolutions

Lines are paralleland lie on top of

one another.

No Solution Lines are paralleland do notintersect

anywhere.

Solution Intersection Two Equations Three Equations


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GEOMETRIC INTERPRETATION

Systems with three variables can be viewed as

planes in 3-D.

Unique Solution Planes intersect

at a single point.

Not possible

Infinitely ManySolutions

Planes intersectat a line or an

entire plane.

No Solution Planes share nocommon

intersection.

Solution Intersection Two Equations Three Equations

linear systems - ch. 1.1, 1.2, 1.3 (part)

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