ALGEBRA 2 X



ALGEBRA 2X

Mr. 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 |ASSIGNMENT |

|1 |Solving by Graphing (and Tables?) |3.1 p.186-189 #1-13, 35-37, 45, 48 |

| |Classifying Linear Systems / # of Solutions | |

|2 |Solving Algebraically (Substitution and Elimination), *Recognizing |3.2 p.194-197 #1-13, 34, 42-44 |

| |Infinitely Many Or No Solutions | |

| |P. 195 (#27 in class) | |

|3 |Systems of Linear Inequalities |3.3 p.202-204 #2-6, 29, 34 |

|4 |3-D Linear Graphs |3.6 p. 224-226 #1-3 |

| |Linear Systems in 3 Variables | |

|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 |Worksheet |

|7 |Determinants and Cramer’s Rule |4.4 p.274 #1-11 (use Calc for 10 and 11), 29, 38, 39|

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

|8 |Review Matrix Multiplication |4.5 p.282-285 #1-12 |

| |Finding Matrix Inverses, Solving Systems Using Matrix Inverses | |

|9 |Row Operations and Augmented Matrices for Solving Systems\ |4.6 p.291-293 #1-9 |

| |Calculator only | |

| |p. 291 #10 if time | |

|10 |Review |p. 300-301 #23-34, 37-50 |

|11 |QUIZ #2 | |

| |(no test for this unit) | |

YOU WILL NEED GRAPH PAPER FOR THIS UNIT

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

[pic]

#3 #4

#5 #6

[pic]

Day 2 Solving Systems of Equations Algebraically

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

[pic]

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).

Solve the system of equations using the substitution method:

1. [pic] 2. [pic]

3. [pic] 4. [pic]

Solve the system of equations using the elimination method:

1. [pic] 2. [pic]

3. [pic] 4. [pic]

Shanae mixes feed for various animals at the zoo so that the feed

has 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 _ 50

b. 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?

|To Graph a System of  Inequalities |Graph the following  system of inequalities  |

| |[pic] |

|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 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. |

|[pic] |For the test point (0,0), |

| |0 < 2(0)-3  False |

| |0 [pic](-2/3)0+2 False |

| |Since 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. |

Section 3-3 cont.

[pic]

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:

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

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 = -2

2x – 2y + z = 7

x + y + 2z = -4

Unit 3 Quiz 1 Review Day 5

[pic]

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 = [pic], then the determinant is _____________.

Other ways to denote determinant: [pic] or [pic]. Notice the straight lines. Do not confuse this notation with __________(topic from last unit rhymes with ‘shmapsolute cow view’).

Examples: [pic] [pic]

[pic] [pic] [pic] [pic]

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

So,

and the solution set is

Cramer’s Rule Practice

|1.   |2.   |3.   |

|6x + 2y = -44 |-1x - 5y = -57 |-5x - 9y = 34 |

|-7x + 9y = -96 |4x - 8y = -108 |8x - 6y = -136 |

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|[pic] | | |

|[pic] | | |

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|4.   |5.   |6.   |

|-3x - 2y = 29 |-5x + 4y = 98 |-1x - 7y = -12 |

|4x - 1y = -57 |6x + 9y = -21 |-3x - 8y = 3 |

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Let’s do a 3x3 on the calculator.

1. 2x – y + 2z = 5

-3x + y – z = -1

x – 3y + 3z = 2

[pic] [pic] [pic][pic]

Closure

What is the determinant of [pic]?

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.

[pic]= [pic]

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

Try These Matrix Multiplications (by hand)

[pic] [pic]

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 [pic].

Formula for Inverse

[pic] What does [pic] mean to do?

Use the formula to find the inverse.

Find the determinant first.

1.) A = [pic] 2.) B = [pic] 3.) C = [pic]

Find [pic] Find [pic] Find [pic]

The following does not have an inverse. WHY?

[pic]

Solving Systems Using Inverse Matrices (sounds scary)

[pic]

Example #1

Rewrite the equation as a MATRIX EQUATION:

[pic]

Coefficient Unknown Constant

Matrix Matrix Matrix

A • X = B

To find x and y, find [pic][pic]

So by hand, you have to find [pic]and then multiply by [pic]

[pic]=

[pic]=

SO, [pic] Which means that x = ___ and y = ___

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

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

[pic]

[pic]

Note: [pic] 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, [pic] and [pic]. What is the 3x3 Identity Matrix?

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

Check:

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

Check:

Example #3: Solve the system [pic] 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.

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:

[pic]

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

[pic]

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:

[pic]

[pic]

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 = 15

3x – 2y = 6

Two Special Cases

Case #1 x + y = 5

3x + 3y = 7 Set up the Augmented Matrix[pic]

Multiply 3 times Row 1 [pic]

Row 2 – Row 1 replaces row 2 [pic]

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 – 6x

3x = 2y + ½

Alternative Method

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

[pic] is reduced to [pic]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 = 6.00

b = cost of a pound of bananas 3a + 2b + 2g + 1.05 = 8.48

g = cost of a pound of grapes 4a + 3b + 2g + 1.05 = 10.46

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.

[pic]

_______________ _______________ ____________

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

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

5. Write the matrix [pic]._________________ The determinant for [pic] is________

6. Write the matrix [pic].________________ The determinant for [pic] 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 [pic]

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

D=____________

9. Find the value of [pic]

[pic]=__________

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.

[pic]

________________________

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

[pic]___________

10. Find [pic] if A=[pic]

[pic]=___________________

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

[pic]

11. Write a matrix equation AX=B.

___________________________________

12. Find [pic]. 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.

[pic]

__________________________

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Day 3

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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?

On Calculator enter:

A = [pic] B = [pic]

THEN,

Multiply [pic]=[pic]

[pic]

Matrix Format: [pic]

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?

3-6

Matrices and Graphing Calculator

1. 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.

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