algebra 2 x - tredyffrin/easttown school district€¦ · web viewalgebra 2x unit 3 graphing...
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ALGEBRA 2 XUNIT 3: LINEAR SYSTEMS OF EQUATIONS
*Assignments and lessons subject to change.*Visit my teacher page for updates, completed notes,
& homework solutions.
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 (50 points) 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 (50 Points)
(no test for this unit)Keystone/SAT review
***BRING YOUR GRAPHING CALCULATOR EVERY DAY***
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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
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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?
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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
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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.
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-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 =
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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:
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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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