Chapter 9 systems of equations and inequalities matrices
This presentation is the property of its rightful owner.
Sponsored Links
1 / 74

Chapter 9: Systems of Equations and Inequalities; Matrices PowerPoint PPT Presentation


  • 112 Views
  • Uploaded on
  • Presentation posted in: General

Chapter 9: Systems of Equations and Inequalities; Matrices. 9.1 Systems of Equations. A set of equations is called a system of equations . The solutions must satisfy each equation in the system. A linear equation in n unknowns has the form where the variables are of first-degree.

Download Presentation

Chapter 9: Systems of Equations and Inequalities; Matrices

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Chapter 9 systems of equations and inequalities matrices

Chapter 9: Systems of Equations and Inequalities; Matrices


9 1 systems of equations

9.1 Systems of Equations

  • A set of equations is called a system of equations.

  • The solutions must satisfy each equation in the system.

  • A linear equation in n unknowns has the form

    where the variables are of first-degree.

  • If all equations in a system are linear, the system is a system of linear equations, or a linear system.


9 1 linear system in two variables

9.1 Linear System in Two Variables

  • Three possible solutions to a linear system in two variables:

    • One solution: coordinates of a point,

    • No solutions: inconsistent case,

    • Infinitely many solutions: dependent case.


Chapter 9 systems of equations and inequalities matrices

9.1

  • Characteristics of a system of two linear equations in two variables.


9 1 substitution method

9.1 Substitution Method

ExampleSolve the system.

Solution

(1)

(2)

Solve (2) for y.

Substitute y = x + 3 in (1).

Solve for x.

Substitute x = 1 in y = x + 3.

Solution set: {(1, 4)}


9 1 solving a linear system in two variables graphically

9.1 Solving a Linear System in Two Variables Graphically

ExampleSolve the system graphically.

SolutionSolve (1) and (2) fory.

(1)

(2)


Chapter 9 systems of equations and inequalities matrices

  • Solve using the method of graphing.

    x2 + y2 = 25

    x2 + y = 19


Solving a nonlinear system of equations

Solving a Nonlinear System of Equations

ExampleSolve the system.

SolutionChoose the simpler equation, (2), and

solve for y since x is squared in (1).

Substitute for y into (1) .

(1)

(2)


Solving a nonlinear system of equations1

Solving a Nonlinear System of Equations

Substitute these values for x into (3).

The solution set is


9 2 elimination method

9.2 Elimination Method

ExampleSolve the system.

SolutionTo eliminate x, multiply (1) by –2 and (2)

by 3 and add the resulting equations.

(1)

(2)

(3)

(4)


9 2 elimination method1

9.2 Elimination Method

Substitute 2 for y in (1) or (2).

The solution set is {(3, 2)}.

Consistent System


Solving an inconsistent system

Solving an Inconsistent System

ExampleSolve the system.

SolutionEliminate x by multiplying (1) by 2 and

adding the result to (2).

Solution set is .

(1)

(2)

Inconsistent System


Solving a system with dependent equations

Solving a System with Dependent Equations

ExampleSolve the system.

SolutionEliminate x by multiplying (1) by 2 and adding the

result to (2).

Consistent & Dependent System

Solution set is all R’s.

(1)

(2)


Chapter 9 systems of equations and inequalities matrices

9.3 Graphing

Systems

of

Inequalities


Chapter 9 systems of equations and inequalities matrices

Look at the two graphs. Determine the following:

A.The equation of each line.

B.How the graphs are similar.

C.How the graphs are different.

  • The equation of each line is y = x + 3.

  • The lines in each graph are the same and represent all of the solutions to the equation y = x + 3.

  • The graph on the right is shaded above the line and this means that all of these points are solutions as well.


Chapter 9 systems of equations and inequalities matrices

Inequalities with Greater Than

Point:(-4, 5)

Pick a point from the shaded region and test that point in the equation y = x + 3.

This is incorrect. Five is greater than or equal to negative 1.

If a solid line is used, then the equation would be 5 ≥ -1.

If a dashed line is used, then the equation would be 5 > -1.

The area above the line is shaded.


Chapter 9 systems of equations and inequalities matrices

Inequalities with Less Than

Point:(1, -3)

Pick a point from the shaded region and test that point in the equation y = -x + 4.

This is incorrect. Negative three is less than or equal to 3.

If a solid line is used, then the equation would be -3 ≤ 3.

If a dashed line is used, then the equation would be -3 < 3.

The area below the line is shaded.


Chapter 9 systems of equations and inequalities matrices

Graphing Linear Inequalities

  • Write the inequality in slope-intercept form.

  • Use the slope and y-intercept to plot two points.

  • Draw in the line. Use a solid line for less than or equal to () or greater than or equal to (≥). Use a dashed line for less than (<) or greater than (>).

  • Pick a point above the line or below the line. Test that point in the inequality. If it makes the inequality true, then shade the region that contains that point. If the point does not make the inequality true, shade the region on the other side of the line.

  • Systems of inequalities – Follow steps 1-4 for each inequality. Find the region where the solutions to the two inequalities would overlap and this is the region that should be shaded.


Chapter 9 systems of equations and inequalities matrices

Example

y

x

Graph the following linear system of inequalities.

Use the slope and y-intercept to plot two points for the first inequality.

Draw in the line. For  use a solid line.

Pick a point and test it in the inequality. Shade the appropriate region.


Chapter 9 systems of equations and inequalities matrices

Example

y

x

Graph the following linear system of inequalities.

The region above the line should be shaded.

Now do the same for the second inequality.


Chapter 9 systems of equations and inequalities matrices

Example

y

x

Graph the following linear system of inequalities.

Use the slope and y-intercept to plot two points for the second inequality.

Draw in the line. For < use a dashed line.

Pick a point and test it in the inequality. Shade the appropriate region.


Chapter 9 systems of equations and inequalities matrices

Example

y

x

Graph the following linear system of inequalities.

The region below the line should be shaded.


Chapter 9 systems of equations and inequalities matrices

Example

y

x

Graph the following linear system of inequalities.

The solution to this system of inequalities is the region where the solutions to each inequality overlap. This is the region above or to the left of the green line and below or to the left of the blue line.

Shade in that region.


Chapter 9 systems of equations and inequalities matrices

You Try It

Graph the following linear systems of inequalities.


Chapter 9 systems of equations and inequalities matrices

Problem 1

y

x

Use the slope and y-intercept to plot two points for the first inequality.

Draw in the line.

Shade in the appropriate region.


Chapter 9 systems of equations and inequalities matrices

Problem 1

y

x

Use the slope and y-intercept to plot two points for the second inequality.

Draw in the line.

Shade in the appropriate region.


Chapter 9 systems of equations and inequalities matrices

Problem 1

y

x

The final solution is the region where the two shaded areas overlap (purple region).


Chapter 9 systems of equations and inequalities matrices

Sect. 9.4 Linear Programming

Goal 1 Find Maximum and Minimum values of a function

Goal 2 Solve Real World Problems with Linear Programming


Chapter 9 systems of equations and inequalities matrices

When graphing a System of Linear Inequalities

* Each linear inequality is called a Constraint

* The intersection of the graphs is called the Feasible Region

* When the graphs of the constraints is a polygonal region, we say the region is Bounded.


Chapter 9 systems of equations and inequalities matrices

Sometimes it is necessary to find the Maximum or Minimum values that a linear function has for the points in a feasible region.

The Maximum or Minimum value of a related function Always occurs at one of the Vertices of the Feasible Region.


Chapter 9 systems of equations and inequalities matrices

Example 1

y  4

y  - x + 6

y 

y  6x + 4

Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the Maximum and Minimum values of the function f(x, y) = 3x + 2y for this polygonal region.


Chapter 9 systems of equations and inequalities matrices

y  4

y  - x + 6

y 

y  6x + 4

The polygon formed is a quadrilateral with vertices at (0, 4), (2, 4), (5, 1), and (- 1, - 2).


Chapter 9 systems of equations and inequalities matrices

Use a table to find the maximum and minimum values of the function.

The maximum value is 17 at (5, 1). The Minimum value is – 7 at (- 1, - 2).


Chapter 9 systems of equations and inequalities matrices

Example 1

Example 2

y  - 3

x  - 2

y 

Bounded Region

Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the Maximum and Minimum values of the function f(x, y) = 2x - 5y for this polygonal region.


Chapter 9 systems of equations and inequalities matrices

The polygon formed is a triangle with vertices at (- 2, 12), (- 2, - 3), and (4, - 3)

The Maximum Value is 23 at (4, - 3). The Minimum Value is – 64 at (- 2, 12).


Chapter 9 systems of equations and inequalities matrices

Sometimes a system of Inequalities forms a region that is not a closed polygon.

In this case, the region is said to be Unbounded.


Chapter 9 systems of equations and inequalities matrices

Example 3

Unbounded Region

Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the Maximum and Minimum values of the function f(x, y) = 4y – 3x for this region.

y + 3x  4

y  - 3x – 4

y  8 + x


Chapter 9 systems of equations and inequalities matrices

There are only two points of intersection (- 1, 7) and (- 3, 5). This is an Unbounded Region.

The Maximum Value is 31 at (- 1, 7). There is no minimum value since there are other points in the solution that produce lesser values. Since the region is Unbounded, f(x, y) has no minimum value.


Chapter 9 systems of equations and inequalities matrices

Linear Programming Procedures.

Step 1: Define the Variables

Step 2: Write a system of Inequalities

Step 3: Graph the System of Inequalities

Step 4: Find the coordinates of the vertices of the feasible region.

Step 5: Write a function to be maximized or minimized.

Step 6: Substitute the coordinates of the vertices into the function.

Step 7: Select the greatest or least result. Answer the problem.


Chapter 9 systems of equations and inequalities matrices

Example 4

Ingrid is planning to start a home-based business. She will be baking decorated cakes and specialty pies. She estimates that a decorated cake will take 75 minutes to prepare and a specialty pie will take 30 minutes to prepare. She plans to work no more than 40 hours per week and does not want to make more than 60 pies in any one week. If she plans to charge $34 for a cake and $16 for a pie, find a combination of cakes and pies that will maximize her income for a week.


Chapter 9 systems of equations and inequalities matrices

Step 1: Define the Variables

C = number of cakes

P = number of pies

Step 2: Write a system of Inequalities

Since number of baked items can’t be negative, c and p must be nonnegative

c  0

p  0

A cake takes 75 minutes and a pie 30 minutes. There are 40 hours per week available.

75c + 30p  240040 hours = 2400 min.

She does not want to make more than 60 pies each week

p  60


Chapter 9 systems of equations and inequalities matrices

Step 3: Graph the system of Inequalities

cakes

Pies


Chapter 9 systems of equations and inequalities matrices

Step 4: Find the Coordinates of the vertices of the feasible region.

The vertices of the feasible region are (0, 0), (0, 32), (60, 8), and (60, 0).

Step 5: Write a function to be maximized or minimized.

The function that describes the income is:

f(p, c) = 16p + 34c


Chapter 9 systems of equations and inequalities matrices

Step 6: Substitute the coordinates of the vertices into function.

Step 7: Select the Greatest or Least result. Answer the Problem.

The maximum value of the function is 1232 at (60, 8). This means that the maximum income is $1232 when Ingrid makes 60 pies and 8 cakes per week.


Section 9 5 solving systems using matrices

Section 9.5Solving Systems using Matrices

  • What is a Matrix? [ ]

  • Augmented Matrices

  • Solving Systems of 3 Equations

  • Inconsistent & Dependent Systems


Concept a matrix

Concept:A Matrix

  • Any rectangular array of numbers arranged in rows and columns, written within brackets

  • Examples:


Chapter 9 systems of equations and inequalities matrices

  • Matrix Row Transformations

    • Streamlined use of echelon methods


9 5 solution of linear systems by row transformations

9.5 Solution of Linear Systems by Row Transformations

  • This is called an augmented matrix where each member of the array is called an element or entry.

  • The rows of an augmented matrix can be treated just like the equations of a linear system.


Concept augmented matrices

Concept:Augmented Matrices

  • Are used to solve systems of linear equations:

    • Arrange equations in simplified standard form

    • Put all coefficients into a 2x3 or 3x4 augmented matrix


Reduced row echelon method with the graphing calculator

Reduced Row Echelon Method with the Graphing Calculator

ExampleSolve the system.

SolutionThe augmented matrix of the system is

shown below.


9 5 reduced row echelon method with the graphing calculator

9.5 Reduced Row Echelon Method with the Graphing Calculator

Using the rref command we obtain the row reduced

echelon form.


9 5 solving a system with no solutions

9.5Solving a System with No Solutions

ExampleShow that the following system is

inconsistent.

SolutionThe augmented matrix of the system is


9 5 solving a system with no solutions1

9.5 Solving a System with No Solutions

The final row indicates that the system is

inconsistent and has solution set .


9 5 solving a system with dependent equations

9.5 Solving a System with Dependent Equations

ExampleShow that the system has dependent

equations. Express the general solution using an

arbitrary variable.

SolutionThe augmented matrix is


9 5 solving a system with dependent equations1

9.5 Solving a System with Dependent Equations

The final row of 0s indicates that the system has

dependent equations. The first two rows represent

the system


9 5 solving a system with dependent equations2

9.5 Solving a System with Dependent Equations

Solving for y we get

Substitute this result into the expression to find x.

Solution set written with z arbitrary:


Matrix algebra basics

Matrix Algebra Basics

9.6


Chapter 9 systems of equations and inequalities matrices

Algebra


Matrix

Matrix

A matrix is any doubly subscripted array of elements arranged in rows and columns.


Row matrix

Row Matrix

[1 x n] matrix


Column matrix

Column Matrix

[m x 1] matrix


Square matrix

Square Matrix

Same number of rows and columns


Chapter 9 systems of equations and inequalities matrices

The Identity


Identity matrix

Identity Matrix

Square matrix with ones on the diagonal and zeros elsewhere.


Matrix addition and subtraction

Matrix Addition and Subtraction

A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B

 is defined by:

Note: all three matrices are of the same dimension


Addition

Addition

If

and

then


Matrix addition example

Matrix Addition Example


Matrix subtraction

Matrix Subtraction

C = A - B

Is defined by


Matrix multiplication

Matrix Multiplication

Matrices A and B have these dimensions:

[r x c] and [s x d]


Matrix multiplication1

Matrix Multiplication

Matrices A and B can be multiplied if:

[r x c] and [s x d]

c = s


Matrix multiplication2

Matrix Multiplication

The resulting matrix will have the dimensions:

[r x c] and [s x d]

r x d


Computation a x b c

Computation: A x B = C

[2 x 2]

[2 x 3]

[2 x 3]


Computation a x b c1

Computation: A x B = C

[3 x 2]

[2 x 3]

A and B can be multiplied

[3 x 3]


Computation a x b c2

Computation: A x B = C

[3 x 2]

[2 x 3]

Result is 3 x 3

[3 x 3]


  • Login