algebra 2 x - tredyffrin/easttown school web viewday topic assignment 1 solving by graphing (and...

ALGEBRA 2XMr. Rives

UNIT 3: LINEAR SYSTEMS OF EQUATIONS

You must SHOW WORK/SETUP for credit (mental math is irrelevant, I need to see your work); FOLLOW

DIRECTIONS

Name_____________________________________DAY TOPIC ASSIGNMENT1 Solving by Graphing (and Tables?)

Classifying Linear Systems / # of Solutions3.1 p.186-189 #1-13, 35-37, 45, 48

2 Solving Algebraically (Substitution and Elimination), *Recognizing Infinitely Many Or No SolutionsP. 195 (#27 in class)

3.2 p.194-197 #1-13, 34, 42-44

3 Systems of Linear Inequalities 3.3 p.202-204 #2-6, 29, 344 3-D Linear Graphs

Linear Systems in 3 Variables3.6 p. 224-226 #1-3

5 Review – start review homework in class; highlight word problems

p.232-235 #1, 3, 5, 6-23, 24-25, 39-48

6 QUIZ #1 Worksheet7 Determinants and Cramer’s Rule

(2x2 by hand and calculator, 3x3 by calculator only)

4.4 p.274 #1-11 (use Calc for 10 and 11), 29, 38, 39

8 Review Matrix MultiplicationFinding Matrix Inverses, Solving Systems Using Matrix Inverses

4.5 p.282-285 #1-12

9 Row Operations and Augmented Matrices for Solving Systems\Calculator onlyp. 291 #10 if time

4.6 p.291-293 #1-9

10 Review p. 300-301 #23-34, 37-5011 QUIZ #2

(no test for this unit)

YOU WILL NEED GRAPH PAPER FOR THIS UNIT

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Algebra 2X Unit 3 Graphing Systems of Linear Equations – Day 1

A system of equations is a set of 2 or more equations containing 2 or more variables.

The solution to a system of equations is an ordered pair where the graphs intersect. (You are looking for the point, or points, that the equations have in common.)

A system with exactly _______________solution(s) is described as consistent and independent.

A system with _________________ _______ solution(s) is consistent and dependent. (How does this occur?)

A system with ___________________ is inconsistent. (How does this occur?)

#1 #2

#3 #4

#5 #6

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Day 2 Solving Systems of Equations Algebraically

1. Solve the system by graphing. Use any method, x and y intercepts work well here.

2. Fill in the blanks:

In an Inconsistent System, the lines are _________________, and there is/are ____________ solution(s).

In an Independent System, the lines are _________________, and there is/are ____________ solution(s).

In a Dependent System, the lines are _________________, and there is/are ____________ solution(s).

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Solve the system of equations using the substitution method:

1. 2.

3. 4.

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Solve the system of equations using the elimination method:

1. 2.

3. 4.

Shanae mixes feed for various animals at the zoo so that the feedhas the right amount of protein. Feed X is 18% protein. Feed Y is 10%protein. Use this data for Exercises 1–2.

1. How much of each feed should Shanae mix to get 50 lb of feed that is 15% protein?a. Write a linear system of equations (you need 2).___________________________________0. 10

.1 5 _ 50b. Solve the system. How much of each feed should she mix? SHOW METHOD HERE

Closure

When is the substitution method more useful to solve a system of linear equations?

What does inconsistent mean in reference to the solution of a linear system of equations? What would the graph look like in general?

An identity such as 7 = 7 is always true and indicates how many solutions?

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To Graph a System of  Inequalities

1.  Graph each inequality separately. 

2.  The solution to the system, will be the ________ where the shadings from each inequality overlap.

Use the grid below.

Graph the following  system of inequalities 

  Graph each inequality as if it was stated in "y=" form.

  If the inequality is < or >, then dotted If the inequality is < or >, then solid

  Choose a test point to determine which side of the line needs to be shaded or do it intuitively.

For the test point (0,0),

0 < 2(0)-3  False

0 (-2/3)0+2 FalseSince both equations were false, shading occurred on the other side of the line, not

covering the point. The solution, S, is where the two shadings overlap one another.

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Day 3

Section 3-3 cont.

Application

Lauren wants to paint no more than 70 plates for a local art fair. It costs her at least $50 plus $2 per plate to create red plates and $3 per plate to create gold plates. She wants to spend no more than $215. Write a system of inequalities that can be used to determine the number of each plate Lauren can create.

Let x = # of red plates

Let y = ____________

Inequalities:

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3.5/3.6 Linear Equations in Three Dimensions/Variables Day 4

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Finish solving here:

Add equation 1 to equation 3. What happened to ‘y?’

Why did we multiply equation 1 by 2?

Based on the diagram, how many solutions are represented?

Based on the diagram, how many solutions are represented?

Based on the diagram, how many solutions are represented?

Solve the system using eliminations to create a system of 2 equations with 2 variables. Solve that system using the methods we have used in this unit. Express your answer as an “ordered triple”.

Example1: x + 2y – 3z = -22x – 2y + z = 7x + y + 2z = -4

Unit 3 Quiz 1 Review Day 5

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3-6

Day 7 Determinants and Cramer’s Rule

All square matrices have a determinant. This value is used to help solve systems of equations.

If A = , then the determinant is _____________.

Other ways to denote determinant: or . Notice the straight lines. Do not

confuse this notation with __________(topic from last unit rhymes with ‘shmapsolute cow view’).

Examples:

Cramer’s Rule

Let’s use this method on a specific example.

To solve 2x – 9y = 9 = (2)(3) – (-9)(6) = 6 + 54 = 60 6x + 3y = 7

x coefficients y coefficients

= (9)(3) – (-9)(7) = 27 + 63 = 90

constants y coefficients

= (2)(7) – (9)(6) = 14 – 54 = -

40

x coefficients constants

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So,

and the solution set is

Cramer’s Rule Practice1.   6x + 2y = -44

-7x + 9y = -962.   -1x - 5y = -57

4x - 8y = -1083.   -5x - 9y = 34

8x - 6y = -136

4.   -3x - 2y = 294x - 1y = -57

5.   -5x + 4y = 986x + 9y = -21

6.   -1x - 7y = -12-3x - 8y = 3

Let’s do a 3x3 on the calculator.

1. 2x – y + 2z = 5Page 13 of 23

Matrices and Graphing Calculator1. Get to the Matrx Menu (some may have a MATRX button, others may need to hit 2nd, x -1 ). 2. Move over to the left to the Edit column. 3. Enter your matrix A as a 3x3 matrix, and enter each element of the matrix. 4. Quit out of the menu.

-3x + y – z = -1 x – 3y + 3z = 2

Closure

What is the determinant of ?

When trying to solve a system of equations using Cramer’s Rule, what do you think a determinant of zero indicates about the solution?

Day 8 Solving Systems Using Matrix Inverses (Review Multiplication)

First, determine the dimensions of the matrices.

=

2 x 2 2 x 3 Do you remember what the dimensions tell you?

Try These Matrix Multiplications (by hand)

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Inverse of a Matrix

Will only work with square matrices. If matrices are inverses of each other, they must be the same size. If A and B are inverse matrices then:

AB = I and BA = I Inverse of matrix A is denoted by .

Formula for Inverse

What does mean to

do?

Use the formula to find the inverse.Find the determinant first.

1.) A = 2.) B = 3.) C =

Find Find Find

The following does not have an inverse. WHY?

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Solving Systems Using Inverse Matrices (sounds scary)

Example #1

Rewrite the equation as a MATRIX EQUATION:

Coefficient Unknown Constant Matrix Matrix Matrix

A • X = B

To find x and y, find

So by hand, you have to find and then multiply by

=

=

SO, Which means that x = ___ and y = ___

Why does this work? Let’s solve a simple equation with inverses.

3x = 9 (multiply by inverse instead of dividing by 3)

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On Calculator enter:

A = B =

THEN,

Multiply =

Note: which is known as the (2 x 2) ______________ matrix

Multiplying a matrix by the identify matrix is like multiplying a number by ____.It does not change it at all!

So, and . What is the 3x3 Identity Matrix?

Example #1: Solve the system by using the inverse of the coefficient matrix.

Check:

Example #2: Solve the system by using the inverse of the coefficient matrix.

Check:

Example #3: Solve the system by using the inverse of the coefficient matrix.

Closure: Fill in the blanks to complete the steps for solving a system using matrices.

Step 1: First I need to write the equations in ______________________. Matrix A is a __ x __ matrix made up of the variables. Matrix B is a __ x __ matrix made up of the constants.

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Step 2: Find the _________________ of matrix _____. The first step to find this is to first find the _______________. The next step is to multiply by __________.

The 3rd step is to make the __________. The final step is to actually multiply.

Step 3: You now need to multiply _____ by _____. Horrrrrrray!

Day 9 Row Operations and Augmented Matrices

In practice, the most common procedure is a combination of row multiplication and row addition. Thinking back to solving two-equation linear systems by addition, you most often had to multiply one row by some number before you added it to the other row. For instance, given:

...you would multiply the first row by 2 before adding it to the second row:

The Goal: To alter the matrix so the first two columns represent the identity matrix and the last column contains the solutions to x and y. Like this:

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Matrix Format:

Let’s try this one together:

3x + 2y = 0 (Be sure to get the x and y on the same side of the equation.)y = -6x + 9

Try this one on your own:3x + y = 153x – 2y = 6

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Two Special CasesCase #1 x + y = 5

3x + 3y = 7 Set up the Augmented Matrix

Multiply 3 times Row 1

Row 2 – Row 1 replaces row 2

The second row translates to 0 + 0 = 8 which is a contradiction. The system is inconsistent – no solution.

Try to see what happens in special case #2.

Case #2 -4y = 1 – 6x3x = 2y + ½

Alternative Method

Instead of trying to get the identity in the first 2 rows, look for a triangle of zeros.

is reduced to The last line indicates 4z = 4 so z =1.

Using substitution yields y = 3. Try solving for x____________________.

Graphing Calculator Example: rref (You’re gonna love it.)

The system of equations represents the costs of three fruit baskets.a = cost of a pound of apples 2a + 2b +g + 1.05 = 6.00b = cost of a pound of bananas 3a + 2b + 2g + 1.05 = 8.48g = cost of a pound of grapes 4a + 3b + 2g + 1.05 = 10.46

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Write the augmented matrix. Enter it into your calculator. Find the cost of a pound of each fruit. Name______________________________Matrices and Solving Systems Review (Day 10)

Unless instructed to do a problem only on your calculator, you must show all work for each problem. You may use a calculator to check work. Put answers on lines provided.

I. Find the determinant value for each matrix below.

_______________ _______________ ____________

II. For the given system, find the information requested.

4. Write the matrix D.___________________ The determinant for D is________

5. Write the matrix ._________________ The determinant for is________

6. Write the matrix .________________ The determinant for is________

7. Use Cramer’s Rule to find the solution to the system. Write the answer as an ordered pair and simplify any fractions.

_____________________________

III. Use your calculator to find the determinants listed below for the system

8. Find the value of D (not the matrix, the DETERMINANT VALUE).

D=____________

9. Find the value of

=__________

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10. What is the value of z in the solution to the system?_______________

IV. Use your knowledge of matrix multiplication for the problems below.

8. Find the result of the multiplication below. Show your work.

________________________

9. Is it possible to multiply the following two matrices together? Explain your answer.

___________

10. Find if A=

=___________________

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V. Use the system below to answer the next problems.

11. Write a matrix equation AX=B.

___________________________________

12. Find . Show your work.

_________________13. Solve the system for x and y. Write your solution as an ordered pair.

14. Use any method on your calculator to solve the system below. Identify the method used—determinants, inverses or rref.

__________________________

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