Lecture 6.1 6.2 bt
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The document outlines the agenda and exam schedule for a class. It then reviews graphing lines and solving systems of linear equations by graphing or algebraically using substitution or elimination methods. Examples are provided to illustrate both algebraic methods. Larger systems of 3 variables are also demonstrated using substitution. Classwork problems are assigned.
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Lecture 6.1 6.2 bt
The document outlines the agenda and schedule for a math class. It includes collecting projects, returning exams, homework due dates, and topics to be covered from sections 6.1, 6.2. Exam dates are provided for Exam 5 on April 1st and Exam 6 on April 27th, with the final exam on May 4th. Methods for solving systems of linear equations are reviewed, including graphing, substitution, and elimination methods. Examples are worked through for each method. Classwork and homework assignments are listed at the end.
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The document outlines the agenda and schedule for a math class. It includes collecting projects, returning exams, homework due dates, and topics to be covered from sections 6.1, 6.2. Exam dates are provided for Exam 5 on April 1st and Exam 6 on April 27th, with the final exam on May 4th. Methods for solving systems of linear equations are reviewed, including graphing, substitution, and elimination methods. Examples are worked through for each method. Classwork and homework assignments are listed at the end.
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3) Using the elimination method, which involves adding or subtracting equations to eliminate a variable and then solving for the remaining variable.
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This document provides instructions on how to solve radical equations. It begins by defining a radical equation and radicand. It then explains the main steps to solve radical equations: isolate the radical to one side, raise both sides to the same power, and simplify. Several examples are worked through to demonstrate this process. The document emphasizes checking solutions, as raising both sides to an even power can result in extraneous solutions. TI instructions and additional practice problems are also included.
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Here are the solutions to the systems of equations without graphing:
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A system of linear equations in two variables can be solved either graphically or algebraically. Graphically, the solutions are found by drawing the lines corresponding to each equation and finding their point(s) of intersection. Algebraically, the equations are combined to eliminate one variable, resulting in an equation that can be solved for the remaining variable. A system has a single solution if the lines intersect at one point, no solution if the lines are parallel, or infinite solutions if the lines coincide as the same line.
Pair of linear equations
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically, the solutions are found by drawing the lines corresponding to each equation and finding their point(s) of intersection. Algebraically, the equations are combined to eliminate one variable, resulting in an equation that can be solved for the remaining variable. A system has a single solution if the lines intersect at one point, no solution if the lines are parallel, or infinite solutions if the lines coincide as the same line.
Solving systems of Linear Equations
There are three possible solutions to a system of linear equations in two variables:
One solution: the graphs intersect at a single point, giving the solution coordinates.
No solution: the graphs are parallel lines, making the system inconsistent.
Infinitely many solutions: the graphs are the same line, making the equations dependent.
The substitution method for solving systems involves: 1) solving one equation for a variable, 2) substituting into the other equation, 3) solving the new equation, and 4) back-substituting to find the remaining variable.
1.8 linear systems
This document discusses linear systems and methods for solving them. A linear system is a set of linear equations where a solution is an ordered pair that satisfies both equations. Graphically, the solution is the point where the lines intersect. Systems can have one solution, infinitely many solutions if the lines are the same, or no solution if the lines are parallel. The document presents two methods for algebraically solving systems: substitution and elimination. Substitution involves solving one equation for a variable and substituting it into the other equation. Elimination involves adding or subtracting equations to eliminate a variable.
Linear equation in two variables
This project was created by four students - Ananya Gupta, Priya Srivastava, Manisha Negi, and Muskan Sharma from Class IX C at KV OFD Raipur in Dehradun, Uttarakhand. The project discusses linear equations and systems of linear equations, explaining concepts such as slope, y-intercept, dependent and independent equations, and methods for solving systems of linear equations graphically, by substitution, and by elimination.
Sytems of linear equation
This document provides an overview of solving systems of linear equations with two or three variables. It defines key terms like systems, solutions, and consistent/inconsistent equations. It describes both graphical and algebraic methods. The algebraic methods covered are elimination, substitution, and Cramer's rule. Examples are provided to illustrate each method. Elimination and back substitution are also described as methods for solving three variable systems algebraically.
February 18, 2015
The document provides instructions for solving different types of systems of equations through various elimination and substitution methods. It includes examples of systems of equations to solve using addition/subtraction, multiplication, and substitution techniques. It also provides instructions for solving systems of inequalities by graphing and shading the solution regions. Finally, it notes some changes made to the assigned class work problems.
Linear equations
This document provides an overview of linear equations for 8th grade algebra. It defines linear equations as having a constant rate of change. For one variable linear equations, it discusses identifying the different types of solutions (conditional, identity, contradiction) and strategies for solving (isolating the variable). For two variable linear equations, it discusses writing the equation in slope-intercept form, graphing, and determining if an ordered pair is a solution. It then covers systems of linear equations, identifying the different types of systems (independent, inconsistent, dependent) and strategies for solving them (graphing, substitution, elimination).
February 13, 2015
1. The document outlines the day's math lesson which includes reviewing systems of equations solutions, solving 3x3 systems, and completing yesterday's class work.
2. It provides examples and steps for solving systems of equations by graphing, elimination, and substitution. Equations are presented in standard form and slope-intercept form.
3. Solving 3x3 systems is discussed, noting they cannot be graphed since they exist in three dimensions. The substitution method is demonstrated through an example.
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Linear Algebra - systems of equations (week 1).ppt
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Derivative power point
1) The document discusses the tangent line problem, one of the major problems that led to the development of calculus. It involves finding the slope of a tangent line to a curve at a point.
2) The derivative of a function f(x) is defined as the limit of the difference quotient as Δx approaches 0. This limit used to define the slope of a tangent line also defines the fundamental operation of differentiation in calculus.
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3.1 derivative of a function
The document defines the derivative of a function and discusses:
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- How the graph of a function's derivative f' relates to the graph of the original function f - where f' is positive/negative/zero corresponds to parts of f that are increasing/decreasing/at an extremum.
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Carpentry2
Students should learn both carpentry and statistics, as carpentry teaches practical skills like using tools and building things while statistics helps students collect and analyze data to answer important questions. A carpentry class focuses on teaching how to use tools and materials to construct projects, whereas a statistics class focuses on teaching students how to gather and make sense of numerical data to solve problems. Both subjects provide valuable skills, but statistics ensures students can use real-world data to find meaningful answers.
Carpentry
Students should learn both carpentry and statistics, as carpentry teaches practical skills like building and construction while statistics allows students to collect and analyze data to answer important questions. A carpentry class focuses on using tools and techniques for woodworking and construction projects, whereas a statistics class emphasizes using numerical and quantitative analysis to study patterns in data and real-world problems. Both subjects provide valuable skills, but statistics equips students to gather and interpret information to find meaningful answers.
Carpentry
Students should learn both carpentry and statistics, as carpentry teaches practical skills like using tools and building things while statistics helps students collect and analyze data to answer important questions. A carpentry class focuses on teaching how to use tools and materials to construct projects, whereas a statistics class focuses on teaching students how to gather and make sense of numerical data to solve problems. Both skills are useful for students to learn.
I book presentation nfd bt
Brad Thompson discusses the potential benefits and realistic applications of digital textbooks. Some key benefits mentioned include lower production costs, easy collaboration across departments, and the ability to customize content. Thompson notes that digital textbooks do not need to be full-sized like traditional books. Several states have passed legislation promoting the adoption of digital textbooks in order to reduce costs for students. Thompson suggests some potential realistic applications at DTCC could include digital lab manuals, learning communities, and instructors publishing their own textbooks. The article closes with a brief introduction to iBook Author.
Lecture 10.4 bt
The document contains an agenda for a statistics class that includes topics such as the normal distribution, z-scores, and using a z-table to find probabilities. The agenda covers defining population and sample means and standard deviations, analyzing exam score data, continuous probability distributions, the properties of the normal distribution, and how to use a z-table to find probabilities for various z-scores and ranges of z-scores. Examples are provided to demonstrate how to find probabilities and areas under the normal curve. Homework problems are assigned from the textbook.
Lecture 10.1 10.2 bt
The document contains an agenda for a math class which includes collecting projects, returning exams, and discussing remaining schedule and topics. It also lists exam dates. There are sections on introducing statistics, describing how to organize and display data through frequency tables, histograms, and polygons. Examples are provided to demonstrate calculating the mean, median, and mode of sample data sets to characterize the data.
Geo 7.3
This document provides an overview of modern geometry concepts covered in Section 7.3, including constructing tangents and secants to circles, properties of inscribed and circumscribed angles, and constructing circles given triangles or vice versa. Key topics are constructing tangents from external points, properties of intersecting secants and secant-tangent intersections, and summarizing circle properties. Homework problems are assigned from pages 369, 378, and 392 of the textbook.
Lecture 7.3 bt
The document outlines the agenda for today's class, which includes turning in a take home quiz, discussing questions from Friday, and covering sections 7.3. It also lists the exam schedule. The rest of the document provides instructions on applying linear programming, including assigning variables, writing constraints as inequalities, finding the objective function, and optimizing over the feasible region. Examples are given to translate a word problem into the constraints and objective function of a linear programming problem.
Geo 7.2 b
The document discusses modern geometry and includes announcements about upcoming exams and teaching presentations. It covers topics like inscribed angles, the star trek lemma, inscribed semicircles, problem solving, and finding the area of a circle segment. Students are assigned problems from the textbook on these topics for homework.
Geo 7.1 7.2
This document provides an overview of key concepts in modern geometry covered in sections 7.1 and 7.2, including:
- Angle and arc notation for circles
- Properties of circles like needing three points to identify a circle
- Measuring arc length and finding the area of sectors of circles
- Theorems about intersecting chords in a circle and relationships between inscribed and central angles
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It also announces an upcoming exam on chapter 7 and assignments for students to complete practice problems.
Lecture 7.1 to 7.2 bt
This document contains an agenda for a class that includes attendance, questions from previous sections, a quiz, and material from sections 7.1 and 7.2. It also lists the exam schedule. Several examples are provided on graphing systems of inequalities and finding feasible regions. Linear programming is introduced as a way to maximize or minimize objective functions subject to constraints. Examples are worked through to find the minimum and maximum values of objective functions over feasible regions defined by constraint inequalities. Students are assigned related practice problems from their textbook.
Lecture 4.4 bt
The document appears to be a class agenda that includes the following:
- Covering section 4.4 in class
- No MML homework this week
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This document contains notes from a geometry class covering sections 6.4 and 6.5 on trigonometric ratios, the unit circle, solving right triangles, laws of sines and cosines, and solving oblique triangles. It includes announcements about an upcoming exam, examples of solving right triangles using trigonometric ratios, formulas for area of triangles, and homework problems assigned from the sections.
Geo 6.2 6.3 bt
This document discusses geometry topics including proving triangles similar using AAA, SAS, and SSS similarity postulates, volume formulas for conic frustums, properties of inscribed right triangles, mean proportions, indirect measurement techniques, and the midsegment and side splitter theorems. It also announces an upcoming exam and homework assignments covering sections 6.2 and 6.3 that are due on Tuesday.
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This document discusses key concepts in geometry related to ratios, proportions, and similarity. It introduces ratios and proportions, covering verifying proportions using cross multiplication and solving proportions. It then discusses similar shapes, geometric means, and the extreme and mean ratio property. Several constructions are presented for the extreme and mean ratio, including using squares, triangles, and the golden ratio. Homework problems are assigned.
Geo 6.2 6.3 bt
This document discusses geometry topics including proving triangles similar using AAA, SAS, and SSS similarity postulates, volume formulas for conic frustums, properties of inscribed right triangles, mean proportions, indirect measurement techniques, and the midsegment and side splitter theorems. It also announces an upcoming exam and homework assignments covering sections 6.2 and 6.3 that are due on Tuesday.
Geo 6.1 bt
This document discusses various topics in modern geometry including:
- Ratios and proportions, and methods for verifying and solving proportions.
- Geometric mean and finding the geometric mean of two numbers.
- The extreme and mean ratio, and constructions for drawing a point B given the ratio of AB to AC.
- Fibonacci numbers and the ratio of successive numbers.
- The golden ratio φ and its relationship to dividing a line segment in the golden ratio.
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Lecture 4.2 bt
The document outlines an agenda that includes:
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Lecture 6.1 6.2 bt
- 1. Today’s Agenda Attendance / Announcements Collect Projects Return Exams Sections 6.1, 6.2 Quiz Friday
- 2. Exam Schedule Exam 4 (Ch 6,7) Fri 11/15 Exam 5 (Ch 10) Thur 12/5 Final Exam (All) Thur 12/12
- 3. Review: Graphing/Equations of Lines Sections 2.1, 2.2
- 4. Systems of Linear Equations 3x y 9 2x y 1 A solution of a system of linear equations is an ordered pair that satisfies both equations.
- 5. Solve by Graphing 3x y 9 2x y 1 So, the solution of the system is (2, -3)
- 6. Solve by Graphing 2x 4 y 8 y 1 2 x 2 So, the solution of the system is ?
- 7. Solve by Graphing x 2y 2x 4 y 3 6 So, the solution of the system is ?
- 9. Solving Systems Algebraically The Substitution Method • Works well when it is easy to solve for one of the variables • Substitute expression into other equation, Solve. • Plug solved value back into either equation to find solution. • Check the solution in both equations
- 10. The Substitution Method 3x y 6 6x 5 y 23 • What does this tell us about the graphs of the system?
- 11. The Substitution Method 3x y 9 2 y 6x 7 • What does this tell us about the graphs of the system?
- 12. The Substitution Method 4x y 8x 2 y 2 4 • What does this tell us about the graphs of the system?
- 13. The Substitution Method x 2y 2x y 3 5 • What does this tell us about the graphs of the system?
- 14. Solving Systems Algebraically The Elimination Method • We can multiply/divide any linear equation. • We can also add/subtract linear equations together. • Use algebra to eliminate one of the variables, solve for the other. • Check the solution in both equations
- 15. The Elimination Method 5x y 4 3x 2 y 1 Addition Method - Want to get “opposite coefficients” - Then add equations to eliminate variable Subtraction Method - Want to get “same coefficients” - Then subtract equations to eliminate variable
- 16. The Elimination Method 2x 3y 2 2 y 3 x 12 Want to rewrite to line up like terms
- 17. The Elimination Method 2 x 5 y 14 7x 6 y 2
- 18. Applications The sum of two numbers is 70 and their difference is 24. Find the two numbers.
- 19. Solving Larger Systems using Substitution • Use algebra to isolate one of the variables, substitute into the other two. • Solve the two variable system, use Substitution find the first variable • Check solution in all three equations
- 20. The Substitution Method 2x y z 2 x 3 y 2z 1 x y z 2
- 21. The Substitution Method 2x 3 y 2 x y 4z 16 3x 2 y z 7
- 22. The Substitution Method x 2y z 5 2x y z 2x 4 y 2z 2 2
- 23. Classwork / Homework • Page 317 •1 – 15 odd, 19 • Page 329 •9, 39, 41, 45