Download 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
1
2
3
4
5
6
7
8
9
10
11
TOPIC
Solving by Graphing (and Tables?)
Classifying Linear Systems / # of Solutions
Solving Algebraically (Substitution and
Elimination), *Recognizing Infinitely Many
Or No Solutions
P. 195 (#27 in class)
Systems of Linear Inequalities
3-D Linear Graphs
Linear Systems in 3 Variables
Review – start review homework in class;
highlight word problems
QUIZ #1
Determinants and Cramer’s Rule
(2x2 by hand and calculator, 3x3 by calculator
only)
Review Matrix Multiplication
Finding Matrix Inverses, Solving Systems
Using Matrix Inverses
Row Operations and Augmented Matrices for
Solving Systems\
Calculator only
p. 291 #10 if time
ASSIGNMENT
3.1 p.186-189 #1-13, 35-37, 45,
48
3.2 p.194-197 #1-13, 34, 42-44
Review
p. 300-301 #23-34, 37-50
3.3 p.202-204 #2-6, 29, 34
3.6 p. 224-226 #1-3
p.232-235 #1, 3, 5, 6-23, 24-25,
39-48
Worksheet
4.4 p.274 #1-11 (use Calc for 10
and 11), 29, 38, 39
4.5 p.282-285 #1-12
4.6 p.291-293 #1-9
QUIZ #2
(no test for this unit)
YOU WILL NEED GRAPH PAPER FOR THIS UNIT
Page 1 of 22
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
x  y  0

7 y  14 x  42
#2
2  y  x

x  y  4
#3
#4
2 x  y  5

3x  2 y  4
#5
y  6 x

3 x  3 y  0
12  4 y  4

2 x  y  6
#6
2 x  y  3

6 x  9  3 y
Page 2 of 22
Page 3 of 22
Day 2 Solving Systems of Equations Algebraically
1.
Solve the system by graphing. Use any method, x and y intercepts work well here.
2 x  4 y  12

 x  2 y  8
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.
 y  2x  3

 x  y  12
2.
x  6 y  6

y  x  4
3.
x  2 y  5

x  4 y 1
4.
 y  2x  5

4 x  2 y  7
Page 4 of 22
Solve the system of equations using the elimination method:
1.
5 x  3 y  10

x  3y  2
2.
5 x  4 y  13

5 x  2 y  7
3.
4 x  7 y  30

2 x  3 y  2
4.
x  4 y  8

2 x  8 y  16
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?
Page 5 of 22
Day 3
To Graph a System of Inequalities
1. Graph each inequality separately.
Graph the following system
of inequalities
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.
For the test point (0,0),
0 < 2(0)-3 False
0 (-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.
Page 6 of 22
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:
Page 7 of 22
3.5/3.6 Linear Equations in Three Dimensions/Variables
Based on the diagram, how many solutions
are represented?
Based on the diagram,
how many solutions are
represented?
Day 4
Based on the diagram,
how many solutions are
represented?
Add equation 1 to equation 3.
What happened to ‘y?’
Why did we multiply equation 1
by 2?
Finish solving here:
Page 8 of 22
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
Page 9 of 22
3-6
Page 10 of 22
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 =
a b 
 c d  , then the determinant is _____________.


Other ways to denote determinant:
A or
a b
c
d
. Notice the straight lines. Do not
confuse this notation with __________(topic from last unit rhymes with ‘shmapsolute
cow view’).
 7 7 

 1 4 
Examples: A  
A
 2 x 5
 1 5 
D

C
 4 20 



 x 10 
2 0
B

 1 9
B
C
D
Cramer’s Rule
Let’s use this method on a specific example.
To solve
2x – 9y = 9
6x + 3y = 7
D 
2 9
6
x coefficients
Dx 
7
= (2)(3) – (-9)(6) = 6 + 54 = 60
y coefficients
9 9
constants
Dy 
3
3
= (9)(3) – (-9)(7) = 27 + 63 = 90
y coefficients
2 9
6 7
= (2)(7) – (9)(6) = 14 – 54 = -40
Page 11 of 22
x coefficients
So,
x 
90 3

60 2
y 
constants
40 2

60
3
 3 2  
and the solution set is  ,  
 2 3  
Cramer’s Rule Practice
1. 6x + 2y = -44
-7x + 9y = -96
6
D 
2
-7 9
Dx 
Dy 

___ ___
___ ___
___ ___
___ ___
x
Dx
D
y
Dy
2. -1x - 5y = -57
4x - 8y = -108
D 

Dx 

Dy 
___ ___
___ ___
___ ___
___ ___
___ ___
___ ___
3. -5x - 9y = 34
8x - 6y = -136

D 

Dx 

Dy 
___ ___
___ ___
___ ___
___ ___
___ ___
___ ___



D
4. -3x - 2y = 29
4x - 1y = -57
5. -5x + 4y = 98
6x + 9y = -21
6. -1x - 7y = -12
-3x - 8y = 3
Page 12 of 22
Let’s do a 3x3 on the calculator.
1. 2x – y + 2z = 5
-3x + y – z = -1
x – 3y + 3z = 2

D 


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.


 D 
 x 







Dy  




D 
 z 






Closure
 4 16 
?
2
8


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.
 5 3  4 2
 0 1  0 1

 
2x2
2x3
(5 x 4)  (3x0)
1 
=

3  (0 x4)  (1x0)

(5 x 2)  (3x1)
(0 x 2)  (1x1)
(5 x  1)  (3 x3) 


(0 x  1)  (1x3) 


Do you remember what the dimensions tell you?
Page 13 of 22
Try These Matrix Multiplications (by hand)
2 0 


 1 4 3 5 1 4 
 2 0 6 1   3 -2 




5 1 
 1 2 3  1
 0 1 0 

 1
 1 9 9   2
9

10 8 

7
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
1
 Inverse of matrix A is denoted by A .
Formula for Inverse
a b 
Let A  
 then
c
d


A1 
1
A
 d b 
 c a 


What does A mean to do?
Use the formula to find the inverse.
Find the determinant first.
2 0
1 3


1.) A =
1
Find A
2.) B =
Find B
6 3 
9 10 


1
 2 2 

 3 4 
3.) C = 
Find C
1
The following does not have an inverse. WHY?
 4 8 
 2 4 


Page 14 of 22
Solving Systems Using Inverse Matrices (sounds scary)
2x  5y  19
3x  4y  6
Example #1
Rewrite the equation as a MATRIX EQUATION:
2 5   x   19 
3 4   y    6 

  

Coefficient
Matrix
A
Unknown
Matrix
•
To find x and y, find
X
Constant
Matrix
=
 19 
B= 

 6 
THEN,
Multiply
A 1 B
x 
= 
y
 
A 1 B
A 1 and then multiply by B
=
A 1 B
SO,
2 5 
A= 

3 4 
B
So by hand, you have to find
A 1
On Calculator enter:
x  
y   
  
=



Why does this work?
AX  B  A1 AX  A1B
 IX  A1 B
 X  A1 B
Which means that x = ___ and y = ___
Let’s solve a simple equation with inverses.
3x = 9
(multiply by inverse instead of dividing by 3)
Page 15 of 22
1 0
Note: I  
 which is known as the (2 x 2) ______________ matrix
0 1
Multiplying a matrix by the identify matrix is like multiplying a number by ____.
It does not change it at all!
So,
IA  A and AI  A .
Example #1: Solve the system
What is the 3x3 Identity Matrix?
2x  3y  1
2x  y  7
by using the inverse of the coefficient
matrix.
Check:
Example #2: Solve the system
8x  5 y  2
3x  2 y  1
by using the inverse of the coefficient
matrix.
Check:
Example #3: Solve the system
2 x  3 y  3
x  2 y  19
by using the inverse of the coefficient
matrix.
Page 16 of 22
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 _____.
Day 9
Horrrrrrray!
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:
Matrix Format:
...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:
1 0
0 1

x  solution 
y  solution 
Page 17 of 22
Let’s try this one together:
3x + 2y = 0
y = -6x + 9
(Be sure to get the x and y on the same side of the equation.)
Try this one on your own:
3x + y = 15
3x – 2y = 6
Page 18 of 22
Two Special Cases
Case #1
x+y=5

Set up the Augmented Matrix 
3x + 3y = 7
Multiply 3 times Row 1




3 1 1 5  3 3 15
3 3 7   3 3 7 

 

Row 2 – Row 1 replaces row 2
1 1 5
0 0 8


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.
 2 3 1 18 
 3 2 4 22 


1 4 2 20 
is reduced to
 2 3 1 18 
 0 10 2 32  The last line indicates 4z = 4 so z =1.


 0 0 4 4 
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.
Page 19 of 22
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.
 3 4
1.) 

 1 7 
 2 1
2.) 

 4 3
_______________
_______________
 0 3
3.) 

 1 7 
____________
2 x  3 y  9

 4x  y  3
4. Write the matrix D.___________________ The determinant for D is________
II.
For the given system, find the information requested.
5. Write the matrix Dx ._________________ The determinant for Dx is________
6. Write the matrix D y .________________ The determinant for D y 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
 3x  2 y  z  6

 3x  3 y  5 z  7
2 x  4 y  3z  14

8. Find the value of D (not the matrix, the DETERMINANT VALUE).
D=____________
9. Find the value of Dz
Dz =__________
10. What is the value of z in the solution to the system?_______________
Page 20 of 22
IV.
Use your knowledge of matrix multiplication for the problems below.
8. Find the result of the multiplication below. Show your work.
2 1   5 2 0 
 3 1  2 3 2



________________________
9. Is it possible to multiply the following two matrices together? Explain your answer.
2
0

7

1
1 3 4  3
4 0 1   2
8 0 0   1

2 3 1  0
0 0 
1 1 
___________
0 1

1 2 
2 7
10. Find A1 if A= 

1 5 
A1 =___________________
V.
Use the system below to answer the next problems.
Page 21 of 22
 2x  3y  5

3x  2 y  10
11. Write a matrix equation AX=B.
___________________________________
12. Find A1 . 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.
 x yz 4

 3x  3 y  z  7
4 x  2 y  z  7

__________________________
Page 22 of 22