Lesson 3-1: Solving Linear Systems by Graphing
|
|
- Andrew Pierce
- 6 years ago
- Views:
Transcription
1 For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we take two or more linear equations together at the same time. Linear Systems When we consider two or more linear equations together at the same time, it is called a linear system. In other words, it is a system (more than one) of linear equations. Perhaps the most interesting thing to look at with linear systems is where the lines intersect (if they do at all). For instance, if we were running a business and had tracked our costs and expenses, we d likely have come up with an equation that predicted what our expenses will be over time. It would also be likely that we d of tracked our profits (hey, everyone loves to make money right?) and would have an equation that predicted our profits over time. If we graphed those two lines, what do you think the intersection of those lines would mean? It would be where our expenses matched our profits right? That would be what we call our breakeven point. After that point, we d be making money. Before then, we d have been losing money. In a system of linear equations, the intersection of the lines is what we refer to as the solution of the linear system. It is the coordinate point (x, y) of the intersection. How do we find the intersection linear system solution? Today and tomorrow we will be learning three ways of finding the solution to a linear system. To give you an idea of where we re going, here are the three ways: 1. Graph the lines, visually locate the intersection: only approximates the solution. 2. Substitution: solve one equation for a variable, substitute into the other. 3. Combination: add the two equations together to eliminate one variable. We will work on the latter two tomorrow. They are the most precise and are algebraic methods. Today we will work on finding the solution by graphing. Finding the intersection by graphing This is pretty straight-forward. Just graph the two lines and find where they intersect. Read off the coordinates and your done right? Not really. How do you know if those coordinates are right? You are drawing the graph; how accurate were you? What if you are off by a tenth? What can you do to decide if the coordinates you read off are correct? Page 1 of 5
2 What does the intersection of two lines mean? It means the point they both go through. Another way of looking at that is that is the only x value for which both equations have the same y value. Plug x into both equations, you should get the same answer from each. That s our answer! Once we read off the coordinates, just plug them into each equation to verify we get the same answer! Easy! What you will need to do today Today s assignment will require you to be able to do the following: 1. Match a system of linear equations with its graph. 2. Determine the number of solutions the system of linear equations has. 3. Decide if a coordinate pair is a solution for a given system of linear equations. 4. Sketch a graph of a system of linear equations to estimate and check the solution. Find the solution of y = 2x + 2 and y = -2x 2 1. Graph both lines. Looking at the graph, it looks like they cross at (-1, 0). Check it: y 2x 2 y 2x 2 0 2( 1) 2 0 2( 1) Both check out, so (-1, 0) is the solution for y = 2x + 2 and y = -2x 2. Page 2 of 5
3 Find the solution of -x + 5y = 5 and 2x 10y = Graph both lines this time they re in standard form x & y intercepts Hmm, a bit of a problem isn t there. It doesn t look like these two lines intersect. Is there an equation form that would make it very easy to compare the two equations? Yup, the slope-intercept form y = mx + b. Solve both equations for y to get them into slope-intercept form: -x + 5y = 5 y = 1 / 5 x + 1 2x 10y = 30 y = 1 / 5 x 3 Great! We can now see that both have the same slope ( 1 / 5 ) but different y-intercepts. Since they cross the y axis at different points this means they are parallel and will never cross. This system of linear equations does not have a solution. Page 3 of 5
4 Find the solution of -2x + y = 3 and 6x 3y = Graph both lines again, they re in standard form x & y intercepts Ok, this was a trick right? They re the same line aren t they? Yup is there a way we can determine that by looking at the equations? Sure is! What is the line form that allows us to directly compare two equations? The slope-intercept form; put both equations into slope-intercept form: -2x + y = 3 y = 2x + 3 6x 3y = -9 y = 2x + 3 Now it is easy to see they are actually the same line! Next question is how many solutions does this system of linear equations have? The number of solutions is the same as the number of times the lines intersect. An intersection is a point both lines have in common. So, how many points do these two equations / lines have in common? All of them, or put another way an infinite number. This system of linear equations has an infinite number of solutions. Page 4 of 5
5 Summary The number of solutions a system of linear equations can have: 1. One: the equations represent different lines that intersect at one point. 2. None: the equations represent parallel lines. 3. Infinite: the equations represent the same line. How to find the solution of a system of linear equations by graphing: 1. Graph both lines. 3. Plug back into both equations and check. How to verify if the lines are parallel or are the same: 1. Put both equations into slope-intercept form (solve both for y). 2. If they have the same slope but different y-intercepts, they are parallel. 3. If they have the same slope and same y-intercepts, they are the same line. Page 5 of 5
Lesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationChapter 6. Systems of Equations and Inequalities
Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system
More informationSections 8.1 & 8.2 Systems of Linear Equations in Two Variables
Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems
More informationSystems of Linear Equations and Inequalities
Systems of Linear Equations and Inequalities Alex Moore February 4, 017 1 What is a system? Now that we have studied linear equations and linear inequalities, it is time to consider the question, What
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationLinear Functions A linear function is a common function that represents a straight line
This handout will: Define Linear and Quadratic Functions both graphically and algebraically Examine the associated equations and their components. Look at how each component could affect shape graphically
More informationPrecalculus idea: A picture is worth 1,000 words
Six Pillars of Calculus by Lorenzo Sadun Calculus is generally viewed as a difficult subject, with hundreds of formulas to memorize and many applications to the real world. However, almost all of calculus
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationPrecalculus: Linear Equations Practice Problems. Questions. 1. Solve for x when 2 3 x = 1 15 x Solve for x when x 2 + x 5 = 7 10.
Questions. Solve for x when 3 x = 5 x + 3 5.. Solve for x when x + x 5 = 7 0. 3. Solve for x when 0 3 x = x. 4. Is 4 a solution to (y ) + = 3 (3y 4)? 8 5. Solve for x when 4 5 x 3 = 3x +. 6. Solve for
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationSection 1.6. Functions
Section 1.6 Functions Definitions Relation, Domain, Range, and Function The table describes a relationship between the variables x and y. This relationship is also described graphically. x y 3 2 4 1 5
More informationO.K. But what if the chicken didn t have access to a teleporter.
The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function
More informationGraphical Solutions of Linear Systems
Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations
More informationLesson 12: Systems of Linear Equations
Our final lesson involves the study of systems of linear equations. In this lesson, we examine the relationship between two distinct linear equations. Specifically, we are looking for the point where the
More informationSTEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is
EQUATIONS OF LINES 1. Writing Equations of Lines There are many ways to define a line, but for today, let s think of a LINE as a collection of points such that the slope between any two of those points
More informationSect The Slope-Intercept Form
0 Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not
More informationSection 1.x: The Variety of Asymptotic Experiences
calculus sin frontera Section.x: The Variety of Asymptotic Experiences We talked in class about the function y = /x when x is large. Whether you do it with a table x-value y = /x 0 0. 00.0 000.00 or with
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More informationCharacteristics of Linear Functions (pp. 1 of 8)
Characteristics of Linear Functions (pp. 1 of 8) Algebra 2 Parent Function Table Linear Parent Function: x y y = Domain: Range: What patterns do you observe in the table and graph of the linear parent
More information3: Linear Systems. Examples. [1.] Solve. The first equation is in blue; the second is in red. Here's the graph: The solution is ( 0.8,3.4 ).
3: Linear Systems 3-1: Graphing Systems of Equations So far, you've dealt with a single equation at a time or, in the case of absolute value, one after the other. Now it's time to move to multiple equations
More informationUnit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where
More informationAlgebra 2 Summer Review Packet
Algebra Summer Review Packet Welcome to Algebra! Attached you will find the learning targets your teacher thinks you should know BEFORE you come to class in the fall and problems to help you practice these
More informationOne Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x
Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have
More informationUnit 1 Science Models & Graphing
Name: Date: 9/18 Period: Unit 1 Science Models & Graphing Essential Questions: What do scientists mean when they talk about models? How can we get equations from graphs? Objectives Explain why models are
More information8th Grade Common Core Math
8th Grade Common Core Math Booklet 5 Functions Part 2 One of the Main Idea of Functions: Use functions to model relationships between quantities What are functions? Functions are like machines. You give
More informationIf you have completed your extra credit opportunity, please place it on your inbox.
Warm-Up If you have completed your extra credit opportunity, please place it on your inbox. On everyone s desk should be paper and a pencil for notes. We are covering all of Quarter 1 in one day, so we
More informationLesson: Slope. Warm Up. Unit #2: Linear Equations. 2) If f(x) = 7x 5, find the value of the following: f( 2) f(3) f(0)
Warm Up 1) 2) If f(x) = 7x 5, find the value of the following: f( 2) f(3) f(0) Oct 15 10:21 AM Unit #2: Linear Equations Lesson: Slope Oct 15 10:05 AM 1 Students will be able to find the slope Oct 16 12:19
More informationMATH 408N PRACTICE MIDTERM 1
02/0/202 Bormashenko MATH 408N PRACTICE MIDTERM Show your work for all the problems. Good luck! () (a) [5 pts] Solve for x if 2 x+ = 4 x Name: TA session: Writing everything as a power of 2, 2 x+ = (2
More informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationHow much does the cow weigh?
1 GRADE 11 PRE-CALCULUS UNIT G SYSTEMS UNIT NOTES 1. Solving Systems of equation is an important subject. To date you have only learned to solve for one unknown in an equation for example: when does 2x
More informationSolving Systems of Linear Equations
Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes
More informationUNIT 1: Lesson 1 Solving for a Variable
AMS I Final Exam Review Packet Name: UNIT 1: Lesson 1 Solving for a Variable To solve for equations you need to use inverse operations. List the inverse operation(s) you could use while solving Addition:
More informationConceptual Explanations: Simultaneous Equations Distance, rate, and time
Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.
More informationUnit 6. Systems of Linear Equations. 3 weeks
Unit 6 Systems of Linear Equations 3 weeks Unit Content Investigation 1: Solving Systems of Linear Equations (3 days) Investigation 2: Solving Systems of Linear Equations by Substitution (4 days) Investigation
More informationComparing linear and exponential growth
Januar 16, 2009 Comparing Linear and Exponential Growth page 1 Comparing linear and exponential growth How does exponential growth, which we ve been studing this week, compare to linear growth, which we
More informationInstruction. Student Activities Overview and Answer Key
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving systems of linear equations using substitution Common Core Standards Analyze and solve linear equations
More informationStudy Unit 2 : Linear functions Chapter 2 : Sections and 2.6
1 Study Unit 2 : Linear functions Chapter 2 : Sections 2.1 2.4 and 2.6 1. Function Humans = relationships Function = mathematical form of a relationship Temperature and number of ice cream sold Independent
More informationSection 5.4. Ken Ueda
Section 5.4 Ken Ueda Students seem to think that being graded on a curve is a positive thing. I took lasers 101 at Cornell and got a 92 on the exam. The average was a 93. I ended up with a C on the test.
More informationName Class Date. What is the solution to the system? Solve by graphing. Check. x + y = 4. You have a second point (4, 0), which is the x-intercept.
6-1 Reteaching Graphing is useful for solving a system of equations. Graph both equations and look for a point of intersection, which is the solution of that system. If there is no point of intersection,
More informationGoing from graphic solutions to algebraic
Going from graphic solutions to algebraic 2 variables: Graph constraints Identify corner points of feasible area Find which corner point has best objective value More variables: Think about constraints
More informationInfinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.
Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions
More informationPre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of
Chapter 1 Beginning at the Very Beginning: Pre-Pre-Calculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope
More informationMA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.
MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationSummer Review. For Students Entering. Algebra 2 & Analysis
Lawrence High School Math Department Summer Review For Students Entering Algebra 2 & Analysis Fraction Rules: Operation Explanation Example Multiply Fractions Multiply both numerators and denominators
More informationRational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.
Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator
More informationMATH 1130 Exam 1 Review Sheet
MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
More informationMath Fundamentals for Statistics I (Math 52) Unit 7: Connections (Graphs, Equations and Inequalities)
Math Fundamentals for Statistics I (Math 52) Unit 7: Connections (Graphs, Equations and Inequalities) By Scott Fallstrom and Brent Pickett The How and Whys Guys This work is licensed under a Creative Commons
More informationChapter 4. Systems of Linear Equations; Matrices
Chapter 4 Systems of Linear Equations; Matrices Section 1 Review: Sys of Linear Eg in Two Var Section 2 Sys of Linear Eq and Aug Matr Section 3 Gauss-Jordan Elimination Section 4 Matrices: Basic Operations
More informationSection 1.4. Meaning of Slope for Equations, Graphs, and Tables
Section 1.4 Meaning of Slope for Equations, Graphs, and Tables Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution Create a table using
More informationLecture 11: Extrema. Nathan Pflueger. 2 October 2013
Lecture 11: Extrema Nathan Pflueger 2 October 201 1 Introduction In this lecture we begin to consider the notion of extrema of functions on chosen intervals. This discussion will continue in the lectures
More informationMath 4: Advanced Algebra Ms. Sheppard-Brick A Quiz Review Learning Targets
5A Quiz Review Learning Targets 4.4 5.5 Key Facts Graphing one-variable inequalities (ex. x < 4 ) o Perform algebra steps to get x alone! If you multiply or divide by a negative number, you must flip the
More informationWe ll start today by learning how to change a repeating decimal into a fraction! Then we will do a review of Unit 1 - half of Unit 3!
Welcome to math! We ll start today by learning how to change a repeating decimal into a fraction! Then we will do a review of Unit 1 - half of Unit 3! So grab a seat where you can focus, and get ready
More information4.1 Solving Systems of Equations Graphically. Draw pictures to represent the possible number of solutions that a linear-quadratic system can have:
4.1 Solving Systems of Equations Graphically Linear- Quadratic A Linear-Quadratic System of Equations is a linear equation and a quadratic equation involving the same two variables. The solution(s) to
More informationPart 1: You are given the following system of two equations: x + 2y = 16 3x 4y = 2
Solving Systems of Equations Algebraically Teacher Notes Comment: As students solve equations throughout this task, have them continue to explain each step using properties of operations or properties
More informationCh. 3 Equations and Inequalities
Ch. 3 Equations and Inequalities 3.1 Solving Linear Equations Graphically There are 2 methods presented in this section for solving linear equations graphically. Normally I would not cover solving linear
More informationLesson 2-6: Graphs of Absolute Value Equations
Where we re headed today Today we re going to take the net graphing step we ll learn how to graph absolute value equations. Here are the three things you are going to need to be able to do: 1. Match an
More informationMath 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations
Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations
More informationExpressions and Equations
Lesson 1 Expressions and Equations Name Use Color Tiles to model each number. Write the perfect square under the radical symbol. Write the square root. 1. 2. 5555 5 = 5 = Using Color Tiles, model each
More informationMath M111: Lecture Notes For Chapter 3
Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following
More informationMATH 320, WEEK 6: Linear Systems, Gaussian Elimination, Coefficient Matrices
MATH 320, WEEK 6: Linear Systems, Gaussian Elimination, Coefficient Matrices We will now switch gears and focus on a branch of mathematics known as linear algebra. There are a few notes worth making before
More informationNext, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.
Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationAnswers for Calculus Review (Extrema and Concavity)
Answers for Calculus Review 4.1-4.4 (Extrema and Concavity) 1. A critical number is a value of the independent variable (a/k/a x) in the domain of the function at which the derivative is zero or undefined.
More information7.1 Solving Linear Systems by Graphing
7.1 Solving Linear Sstems b Graphing Objectives: Learn how to solve a sstem of linear equations b graphing Learn how to model a real-life situation using a sstem of linear equations With an equation, an
More informationPHYSICS LAB: CONSTANT MOTION
PHYSICS LAB: CONSTANT MOTION Introduction Experimentation is fundamental to physics (and all science, for that matter) because it allows us to prove or disprove our hypotheses about how the physical world
More informationParabolas and lines
Parabolas and lines Study the diagram at the right. I have drawn the graph y = x. The vertical line x = 1 is drawn and a number of secants to the parabola are drawn, all centred at x=1. By this I mean
More informationGraphing Systems of Linear Equations
Graphing Systems of Linear Equations Groups of equations, called systems, serve as a model for a wide variety of applications in science and business. In these notes, we will be concerned only with groups
More informationLinear Equations and Functions
Linear Equations and Functions Finding the Slope and Equation of a Line. ans-00-0.... x y y x b c c b Finding Slope.... undefined. 6. - 7. - 8. 0 9. 0. undefined.. 6... 6. (, -) Finding the Equation of
More informationUnit 7: It s in the System
Unit 7: It s in the System Investigation 1: Linear Equations with Two Variables I can convert between standard and slope intercept forms, and graph systems of equations. Solving equations is one of the
More informationReteach 2-3. Graphing Linear Functions. 22 Holt Algebra 2. Name Date Class
-3 Graphing Linear Functions Use intercepts to sketch the graph of the function 3x 6y 1. The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x. 3x 6y 1
More information1.2 Functions & Function Notation
1.2 Functions & Function Notation A relation is any set of ordered pairs. A function is a relation for which every value of the independent variable (the values that can be inputted; the t s; used to call
More information=.55 = = 5.05
MAT1193 4c Definition of derivative With a better understanding of limits we return to idea of the instantaneous velocity or instantaneous rate of change. Remember that in the example of calculating the
More informationWEEK 7 NOTES AND EXERCISES
WEEK 7 NOTES AND EXERCISES RATES OF CHANGE (STRAIGHT LINES) Rates of change are very important in mathematics. Take for example the speed of a car. It is a measure of how far the car travels over a certain
More informationWe are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero
Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.
More informationSect 2.4 Linear Functions
36 Sect 2.4 Linear Functions Objective 1: Graphing Linear Functions Definition A linear function is a function in the form y = f(x) = mx + b where m and b are real numbers. If m 0, then the domain and
More information4 The Cartesian Coordinate System- Pictures of Equations
4 The Cartesian Coordinate System- Pictures of Equations Concepts: The Cartesian Coordinate System Graphs of Equations in Two Variables x-intercepts and y-intercepts Distance in Two Dimensions and the
More informationSystems of Equations and Inequalities. College Algebra
Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system
More informationLesson 6-1: Relations and Functions
I ll bet you think numbers are pretty boring, don t you? I ll bet you think numbers have no life. For instance, numbers don t have relationships do they? And if you had no relationships, life would be
More informationCP Algebra 2. Unit 3B: Polynomials. Name: Period:
CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationCS 301. Lecture 18 Decidable languages. Stephen Checkoway. April 2, 2018
CS 301 Lecture 18 Decidable languages Stephen Checkoway April 2, 2018 1 / 26 Decidable language Recall, a language A is decidable if there is some TM M that 1 recognizes A (i.e., L(M) = A), and 2 halts
More informationSection 4.6 Negative Exponents
Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals
More informationGeometry Summer Assignment 2018
Geometry Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Geometry this year. You are advised to be familiar with each
More information2.4 Graphing Inequalities
.4 Graphing Inequalities Why We Need This Our applications will have associated limiting values - and either we will have to be at least as big as the value or no larger than the value. Why We Need This
More informationUnit 5: Moving Straight Ahead
Unit 5: Moving Straight Ahead Investigation 3 Solving Equations I can recognize problem situations in which two variables have a linear relationship and solve rate of change problems. In the last Investigation,
More information6-4 Solving Special Systems
Warm Up Solve each equation. 1. 2x + 3 = 2x + 4 2. 2(x + 1) = 2x + 2 3. Solve 2y 6x = 10 for y Solve by using any method. 4. y = 3x + 2 2x + y = 7 5. x y = 8 x + y = 4 Know: Solve special systems of linear
More informationTo get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.
Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function
More informationProb and Stats, Sep 23
Prob and Stats, Sep 23 Calculator Scatter Plots and Equations of Lines of Fit Book Sections: 4.1 Essential Questions: How can the calculator help me to produce a scatter plot, and also the equation of
More informationJune If you want, you may scan your assignment and convert it to a.pdf file and it to me.
Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular
More informationPeriod: Date: Lesson 3B: Properties of Dilations and Equations of lines
Name: Period: Date: : Properties of Dilations and Equations of lines Learning Targets I can identify the properties of dilation mentioned as followed: dilation takes a line not passing through the center
More informationSince the two-sided limits exist, so do all one-sided limits. In particular:
SECTION 3.6 IN-SECTION EXERCISES: EXERCISE 1. The Intermediate Value Theorem 1. There are many correct graphs possible. A few are shown below. Since f is continuous on [a, b] and π is between f(a) = 3
More informationP1 Chapter 3 :: Equations and Inequalities
P1 Chapter 3 :: Equations and Inequalities jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 26 th August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationElliptic Curves. Dr. Carmen Bruni. November 4th, University of Waterloo
University of Waterloo November 4th, 2015 Revisit the Congruent Number Problem Congruent Number Problem Determine which positive integers N can be expressed as the area of a right angled triangle with
More informationWater tank. Fortunately there are a couple of objectors. Why is it straight? Shouldn t it be a curve?
Water tank (a) A cylindrical tank contains 800 ml of water. At t=0 (minutes) a hole is punched in the bottom, and water begins to flow out. It takes exactly 100 seconds for the tank to empty. Draw the
More informationAlgebra 2 Honors Unit 1 Review of Algebra 1
Algebra Honors Unit Review of Algebra Day Combining Like Terms and Distributive Property Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents SWBAT combine
More informationOur first case consists of those sequences, which are obtained by adding a constant number d to obtain subsequent elements:
Week 13 Sequences and Series Many images below are excerpts from the multimedia textbook. You can find them there and in your textbook in sections 7.2 and 7.3. We have encountered the first sequences and
More information