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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[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Day 3
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
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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