6-2 Solving Systems by Substitution
|
|
- Morgan Nicholson
- 5 years ago
- Views:
Transcription
1 6-2 Solving Systems by Substitution Warm Up Lesson Presentation Lesson Quiz 1
2 Warm Up Solve each equation for x. 1. y = x y = 3x 4 Simplify each expression. 3. 2(x 5) x = y 3 2x (x + 1) 9 3x
3 Warm Up Continued Evaluate each expression for the given value of x. 5. x + 8 for x = (x 7) for x =10 9
4 Objective Solve linear equations in two variables by substitution.
5 Sometimes it is difficult to identify the exact solution to a system by graphing. In this case, you can use a method called substitution. The goal when using substitution is to reduce the system to one equation that has only one variable. Then you can solve this equation by the methods taught in Chapter 2.
6 Solving Systems of Equations by Substitution Step 1 Solve for one variable in at least one equation, if necessary. Step 2 Step 3 Step 4 Step 5 Substitute the resulting expression into the other equation. Solve that equation to get the value of the first variable. Substitute that value into one of the original equations and solve. Write the values from steps 3 and 4 as an ordered pair, (x, y), and check.
7 Example 1A: Solving a System of Linear Equations by Substitution Solve the system by substitution. y = 3x y = x 2 Step 1 y = 3x y = x 2 Step 2 y = x 2 3x = x 2 Step 3 x x 2x = 2 2x = x = 1 Both equations are solved for y. Substitute 3x for y in the second equation. Solve for x. Subtract x from both sides and then divide by 2.
8 Example 1A Continued Solve the system by substitution. Step 4 y = 3x y = 3( 1) y = 3 Step 5 ( 1, 3) Write one of the original equations. Substitute 1 for x. Write the solution as an ordered pair. Check Substitute ( 1, 3) into both equations in the system. y = 3x y = x 2 3 3( 1)
9 Example 1B: Solving a System of Linear Equations by Substitution Solve the system by substitution. y = x + 1 4x + y = 6 Step 1 y = x + 1 The first equation is solved for y. Step 2 4x + y = 6 4x + (x + 1) = 6 Substitute x + 1 for y in the second equation. 5x + 1 = 6 Simplify. Solve for x. Step Subtract 1 from both sides. 5x = 5 5x = 5 Divide both sides by x = 1
10 Example1B Continued Solve the system by substitution. Step 4 y = x + 1 y = y = 2 Step 5 (1, 2) Write one of the original equations. Substitute 1 for x. Write the solution as an ordered pair. Check Substitute (1, 2) into both equations in the system. y = x + 1 4x + y = (1)
11 Example 1C: Solving a System of Linear Equations by Substitution Solve the system by substitution. x + 2y = 1 x y = 5 Step 1 x + 2y = 1 2y 2y x = 2y 1 Step 2 x y = 5 ( 2y 1) y = 5 3y 1 = 5 Simplify. Solve the first equation for x by subtracting 2y from both sides. Substitute 2y 1 for x in the second equation.
12 Step 3 3y 1 = y = 6 Example 1C Continued 3y = y = 2 Step 4 x y = 5 x ( 2) = 5 x + 2 = x = 3 Step 5 (3, 2) Solve for y. Add 1 to both sides. Divide both sides by 3. Write one of the original equations. Substitute 2 for y. Subtract 2 from both sides. Write the solution as an ordered pair.
13 Check It Out! Example 1a Solve the system by substitution. y = x + 3 y = 2x + 5 Step 1 y = x + 3 y = 2x + 5 Step 2 y = x + 3 2x + 5 = x + 3 Step 3 2x + 5 = x + 3 x 5 x 5 x = 2 Both equations are solved for y. Substitute 2x + 5 for y in the first equation. Solve for x. Subtract x and 5 from both sides.
14 Check It Out! Example 1a Continued Solve the system by substitution. Step 4 y = x + 3 y = y = 1 Write one of the original equations. Substitute 2 for x. Step 5 ( 2, 1) Write the solution as an ordered pair.
15 Check It Out! Example 1b Solve the system by substitution. x = 2y 4 x + 8y = 16 Step 1 x = 2y 4 Step 2 x + 8y = 16 (2y 4) + 8y = 16 The first equation is solved for x. Substitute 2y 4 for x in the second equation. Step 3 10y 4 = 16 Simplify. Then solve for y y = 20 Add 4 to both sides. 10y 20 = Divide both sides by 10. y = 2
16 Check It Out! Example 1b Continued Solve the system by substitution. Step 4 x + 8y = 16 Write one of the original equations. x + 8(2) = 16 Substitute 2 for y. Step 5 (0, 2) x + 16 = x = 0 Simplify. Subtract 16 from both sides. Write the solution as an ordered pair.
17 Check It Out! Example 1c Solve the system by substitution. 2x + y = 4 x + y = 7 Step 1 x + y = 7 y y x = y 7 Step 2 x = y 7 2( y 7) + y = 4 Solve the second equation for x by subtracting y from each side. Substitute y 7 for x in the first equation. 2( y 7) + y = 4 2y 14 + y = 4 Distribute 2.
18 Check It Out! Example 1c Continued Solve the system by substitution. Step 3 2y 14 + y = 4 y 14 = y = 10 y = 10 Combine like terms. Add 14 to each side. Step 4 x + y = 7 Write one of the original equations. x + ( 10) = 7 Substitute 10 for y. x 10 = 7
19 Check It Out! Example 1c Continued Solve the system by substitution. Step 5 x 10 = x = 3 Add 10 to both sides. Step 6 (3, 10) Write the solution as an ordered pair.
20 Sometimes you substitute an expression for a variable that has a coefficient. When solving for the second variable in this situation, you can use the Distributive Property.
21 Caution When you solve one equation for a variable, you must substitute the value or expression into the other original equation, not the one that had just been solved.
22 Example 2: Using the Distributive Property Solve y + 6x = 11 3x + 2y = 5 by substitution. Step 1 y + 6x = 11 6x 6x y = 6x + 11 Step 2 3x + 2y = 5 3x + 2( 6x + 11) = 5 Solve the first equation for y by subtracting 6x from each side. Substitute 6x + 11 for y in the second equation. 3x + 2( 6x + 11) = 5 Distribute 2 to the expression in parenthesis.
23 Example 2 Continued y + 6x = 11 Solve by substitution. 3x + 2y = 5 Step 3 3x + 2( 6x) + 2(11) = 5 3x 12x + 22 = 5 9x + 22 = x = 27 9x = x = 3 Simplify. Solve for x. Subtract 22 from both sides. Divide both sides by 9.
24 Solve Step 4 y + 6x = 11 y + 6(3) = 11 y + 18 = Example 2 Continued y + 6x = 11 3x + 2y = 5 y = 7 by substitution. Write one of the original equations. Substitute 3 for x. Simplify. Subtract 18 from each side. Step 5 (3, 7) Write the solution as an ordered pair.
25 Check It Out! Example 2 Solve 2x + y = 8 3x + 2y = 9 by substitution. Step 1 2x + y = 8 + 2x +2x y = 2x + 8 Step 2 3x + 2y = 9 3x + 2(2x + 8) = 9 Solve the first equation for y by adding 2x to each side. Substitute 2x + 8 for y in the second equation. 3x + 2(2x + 8) = 9 Distribute 2 to the expression in parenthesis.
26 Check It Out! Example 2 Continued 2x + y = 8 Solve by substitution. 3x + 2y = 9 Step 3 3x + 2(2x) + 2(8) = 9 3x + 4x + 16 = 9 7x + 16 = x = 7 7x = x = 1 Simplify. Solve for x. Subtract 16 from both sides. Divide both sides by 7.
27 Solve Check It Out! Example 2 Continued 2x + y = 8 3x + 2y = 9 Step 4 2x + y = 8 2( 1) + y = 8 y + 2 = y = 6 by substitution. Write one of the original equations. Substitute 1 for x. Simplify. Subtract 2 from each side. Step 5 ( 1, 6) Write the solution as an ordered pair.
28 Example 2: Consumer Economics Application Jenna is deciding between two cell-phone plans. The first plan has a $50 sign-up fee and costs $20 per month. The second plan has a $30 sign-up fee and costs $25 per month. After how many months will the total costs be the same? What will the costs be? If Jenna has to sign a one-year contract, which plan will be cheaper? Explain. Write an equation for each option. Let t represent the total amount paid and m represent the number of months.
29 Example 2 Continued Total paid is signup fee plus payment amount for each month. Option 1 t = $50 + $20 m Option 2 t = $30 + $25 m Step 1 t = m t = m Step m = m Both equations are solved for t. Substitute m for t in the second equation.
30 Step m = m Solve for m. Subtract 20m 20m 20m from both sides. 50 = m Subtract 30 from both sides. 20 = 5m Divide both sides by = 5m 5 5 Step 4 t = m t = (4) t = t = 130 Example 2 Continued m = 4 Write one of the original equations. Substitute 4 for m. Simplify.
31 Step 5 (4, 130) Example 2 Continued Write the solution as an ordered pair. In 4 months, the total cost for each option would be the same $130. If Jenna has to sign a one-year contract, which plan will be cheaper? Explain. Option 1: t = (12) = 290 Option 2: t = (12) = 330 Jenna should choose the first plan because it costs $290 for the year and the second plan costs $330.
32 Check It Out! Example 3 One cable television provider has a $60 setup fee and $80 per month, and the second has a $160 equipment fee and $70 per month. a. In how many months will the cost be the same? What will that cost be. Write an equation for each option. Let t represent the total amount paid and m represent the number of months.
33 Check It Out! Example 3 Continued Total paid is fee plus payment amount for each month. Option 1 t = $60 + $80 m Option 2 t = $160 + $70 m Step 1 t = m t = m Both equations are solved for t. Step m = m Substitute m for t in the second equation.
34 Check It Out! Example 3 Continued Step m = m Solve for m. Subtract 70m 70m 70m from both sides m = 160 Subtract 60 from both sides. 10m = 100 Divide both sides by m = 10 Step 4 t = m t = (10) t = t = 860 Write one of the original equations. Substitute 10 for m. Simplify.
35 Check It Out! Example 3 Continued Step 5 (10, 860) Write the solution as an ordered pair. In 10 months, the total cost for each option would be the same, $860. b. If you plan to move in 6 months, which is the cheaper option? Explain. Option 1: t = (6) = 540 Option 2: t = (6) = 580 The first option is cheaper for the first six months.
36 Lesson Quiz: Part I Solve each system by substitution. y = 2x 1. ( 2, 4) x = 6y (1, 2) 3x 2y = x + y = 1 x y = 4
37 Lesson Quiz: Part II 4. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each hour. For how many hours will the total cost for each plumber be the same? How much will that cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the customer hire? Explain. 8 hours; $480; plumber A: plumber A is cheaper for less than 8 hours.
1-3 Variables and Algebraic Expressions. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Warm Up Evaluate. 1. 5(7) 1 2. 7(18 11) 3. 22 + 17 8 + 3 4. 36 + 15(40 35) 5. 3 3 + 7(12 4) Problem of the Day If charged per cut, how much
More information3-6 Lines in the Coordinate Plane
3-6 Lines in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 1. m = 2, x = 3, and
More information1-3 Multiplying and Dividing Real Numbers
Multiplying and Dividing 1-3 Multiplying and Dividing Real Numbers Real Numbers Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 1-3 Add or Subtract 1. 3 8 2 pts 2. - 8 + 12 2 pts 3. 4 (-4) 2
More informationSolving Algebraic Equations
Lesson 4. Solving Algebraic Equations 3 3 3 3 3 8 8 4 Add 3 to both sides. Divide both sides by. 4 gives the solution of the equation 3. Check: Substitute 4 for x into the original equation. 3 4 3 When
More informationUnit 2A: Systems of Equations and Inequalities
Unit A: Systems of Equations and Inequalities In this unit, you will learn how to do the following: Learning Target #1: Creating and Solving Systems of Equations Identify the solution to a system from
More informationWarm Up Simplify each expression. Assume all variables are nonzero.
Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x 6 x 2 x 7 x 4 Factor each expression. 2. y 3 y 3 y 6 4. y 2 1 y 5 y 3 5. x 2 2x 8 (x 4)(x + 2) 6. x 2 5x x(x 5) 7. x
More informationGeometric Sequences. Geometric Sequences. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1
Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Find the value of each expression. 1. 2 5 32 2. 2 5 3. 3 4 81 4. ( 3) 4 81 5. (0.2) 3 0.008 6. 7( 4) 2 112 7. 8. 12( 0.4) 3 0.768 Objectives Recognize
More information7th Grade Math Unit 1 Algebraic Reasoning
7th Grade Math Unit 1 Algebraic Reasoning Name: Period: Common Core State Standards CC.7.NS.1 - Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers;
More informationHelp! I m Broke and I Need A Phone! PASS OBJECTIVES: MATERIALS Teacher PROCEDURES: SUMMARY:
Help! I m Broke and I Need A Phone! PASS OBJECTIVES: Standard 1.1 Translate word phrases and sentences into expressions and equations and vice versa. Standard 2.3 Calculate the slope of a line using a
More informationLesson 8 - Practice Problems
Lesson 8 - Practice Problems Section 8.1: A Case for the Quadratic Formula 1. For each quadratic equation below, show a graph in the space provided and circle the number and type of solution(s) to that
More informationTalk Is Cheap A-E Strand(s): Algebra
A-E Strand(s): Algebra Topic/Expectation A.A.2 Functions b. Represent and interpret functions using graphs, tables, words, and symbols. A.A.3 Linear Functions a. Analyze and identify linear functions of
More informationNote-Taking Guides. How to use these documents for success
1 Note-Taking Guides How to use these documents for success Print all the pages for the module. Open the first lesson on the computer. Fill in the guide as you read. Do the practice problems on notebook
More informationGUIDELINES FOR COMPLETING THE ASSIGNMENT
RAHWAY HIGH SCHOOL MATHEMATICS DEPARTMENT Algebra 1 Summer Assignment packet Summer 2018 Due date: September 7th GUIDELINES FOR COMPLETING THE ASSIGNMENT This packet was created to help you succeed in
More information12-4 Geometric Sequences and Series. Lesson 12 3 quiz Battle of the CST s Lesson Presentation
12-4 Geometric Sequences and Series Lesson 12 3 quiz Battle of the CST s Lesson Presentation Objectives Find terms of a geometric sequence, including geometric means. Find the sums of geometric series.
More information1-3 Square Roots. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
1-3 Square Roots Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Round to the nearest tenth. 1. 3.14 3.1 2. 1.97 2.0 Find each square root. 3. 4 4. 25 Write each fraction in simplest form. 5. 6. Simplify.
More informationModule 11 & 12. Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities
Module 11 & 12 Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities What is a System of Equations? A system of linear equations consists
More information2.4 Solving Linear Equations
2.4 Solving Linear Equations When we are solving equations, we are attempting to isolate the variable in order to determine what specific value that variable has in the given equation. We do this using
More informationMath and 4.5 Practice Quiz
Class: Date: Math 9 4.4 and 4.5 Practice Quiz Short Answer 1. Which graph on this grid has the equation y = 3x? 2. Which graph on this grid has the equation y = x + 3? 1 3. Which equation describes the
More information6-3 Solving Systems by Elimination
Warm Up Simplify each expression. 1. 3x + 2y 5x 2y 2. 5(x y) + 2x + 5y 3. 4y + 6x 3(y + 2x) 4. 2y 4x 2(4y 2x) Write the least common multiple. 5. 3 and 6 6. 4 and 10 7. 6 and 8 8. 2 and 5 Learning Goals
More information10-2 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
10-2 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
More information2.1 Solutions to Exercises
Last edited 9/6/17.1 Solutions to Exercises 1. P(t) = 1700t + 45,000. D(t) = t + 10 5. Timmy will have the amount A(n) given by the linear equation A(n) = 40 n. 7. From the equation, we see that the slope
More information3.5 Write and Graph Equations
.5 Write and Graph Equations of Lines Goal p Find equations of lines. Your Notes VOCABULARY Slope-intercept form Standard form Eample Write an equation of a line from a graph Write an equation of the line
More informationCurve Fitting with Linear Models
1-4 1-4 Curve Fitting with Linear Models Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Write the equation of the line passing through each pair of passing points in slope-intercept form. 1.
More informationSubtraction Understand Subtraction on a Number Line Using a number line let s demonstrate the subtraction process using the problem 7 5.
Objective 1 Subtraction Understand Subtraction on a Number Line Using a number line let s demonstrate the subtraction process using the problem 7 5. -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 Using the number line
More informationModule 11 & 12. Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities
Module 11 & 12 Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities What is a System of Equations? A system of linear equations consists
More informationExponents. Reteach. Write each expression in exponential form (0.4)
9-1 Exponents You can write a number in exponential form to show repeated multiplication. A number written in exponential form has a base and an exponent. The exponent tells you how many times a number,
More informationAlg. 1 Unit 3 Notes Unit 3 Day 1: Represent Relations and Functions (O.C. 1-5)
Alg. 1 Unit 3 Notes Unit 3 Day 1: Represent Relations and Functions (O.C. 1-5) A. Vocabulary Objectives: SWBAT represent functions Function Function Notation Coordinate Domain Range State the domain, the
More information6.7. Graph Linear Inequalities in Two Variables. Warm Up Lesson Presentation Lesson Quiz
6.7 Graph Linear Inequalities in Two Variables Warm Up Lesson Presentation Lesson Quiz 6.7 Warm-Up Tell whether the ordered pair is a solution of the equation. 1. x + 2y = 4; (2, 1) no 2. 4x + 3y = 22;
More information3.2 Graphs of Linear Equations
3.2 Graphs of Linear Equations Learning Objectives Graph a linear function using an equation. Write equations and graph horizontal and vertical lines. Analyze graphs of linear functions and read conversion
More informationLesson 24 - Exploring Graphical Transformations and Composite Functions
(A) Lesson Objectives a. Review composite functions and how it can be represented numerically, algebraically and graphically. b. Introduce graphical transformations c. Understand that graphical transformations
More informationDLA Review Printable Version
1. In the equation y = 7x + 3, as the value of x decreases by 1, what happens to the value of y?. A cell phone company charges $.00 a month plus an additional $0.10 per call. A competitor charges $10.00
More informationApplication checklist
Application checklist I enclose with this form (please tick the boxes that apply): Photocopy of the photo I.D. you intend to take with you to the assessment (valid passport or UK/EU driving license (full
More information3.1 Start Thinking. 3.1 Warm Up. 3.1 Cumulative Review Warm Up. Consider the equation y x.
3.1 Start Thinking Consider the equation y x. Are there any values of x that you cannot substitute into the equation? If so, what are they? Are there any values of y that you cannot obtain as an answer?
More information2-9 Operations with Complex Numbers
2-9 Operations with Complex Numbers Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Express each number in terms of i. 1. 9i 2. Find each complex conjugate. 3. 4. Find each product. 5. 6. Objective
More informationRegistration assessment application Autumn 2018 sitting
Registration assessment application Autumn 2018 sitting Application checklist I enclose with this form (please tick the boxes that apply): Photocopy of the photo I.D. you intend to take with you to the
More informationEssential Questions. Key Terms. Algebra. Arithmetic Sequence
Linear Equations and Inequalities Introduction Average Rate of Change Coefficient Constant Rate of Change Continuous Discrete Domain End Behaviors Equation Explicit Formula Expression Factor Inequality
More informationCEU Catalog Guide. When you access the CEU catalog it defaults to ALL available CEUs.
CEU Catalog Guide When you access the CEU catalog it defaults to ALL available CEUs. You can see the Title of the CEU, the Certification(s) it will apply to, Topic Code and Credit Hours Below the Title
More informationDecember 05, Ch4.notebook. Objective: To learn how to analyze and write function rules. Warm Up 11/30. Nov 30 7:36 AM
Objective: To learn how to analyze and write function rules. Warm Up 11/30 Nov 30 7:36 AM 1 Objective: To learn how to analyze and write function rules. Warm Up 12/1 Dec 1 7:37 AM 2 Objective: To learn
More information1-6 Order of Operations
1-6 Order of Operations Warm Up Lesson Presentation Lesson Quiz 2 pts 3 pts Bell Quiz 1-6 Find each square root. 1. 25 Write all classifications that apply to each real number. 3. -55 5 pts possible Questions
More informationWelcome back! Sit down and work on the warm up!
Welcome back! Sit down and work on the warm up! 1 Rewrite 4 4 4 4 4 4 4 4 4 using exponents 2 Circle the coefficient and square the constant of the function y=3x+7 3 Rewrite the expression using multiplication:
More informationAnswers Investigation 3
Answers Applications 1. a. 25 shirts would cost $70. You could use a table by trying to find the cost C for every value of n. Thus, the table would reflect values for n = 1, 2, 3,..., 25. You could use
More informationReady to Go On? Skills Intervention 1-1. Exploring Transformations. 2 Holt McDougal Algebra 2. Name Date Class
Lesson - Read to Go n? Skills Intervention Eploring Transformations Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular transformation translation reflection stretch Translating
More informationCh. 5.1: Write Linear Equations in Slope-Intercept Form. Example 1: Write the equation of the line with a slope of 2 and a y-intercept of 5.
Chapter 5 Notes A2 Short Answer 1. Ch. 5.1: Write Linear Equations in Slope-Intercept Form Example 1: Write the equation of the line with a slope of 2 and a y-intercept of 5. Example 2: Write the equation
More information2-2 Adding Integers. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Warm Up Find each absolute value. 1. 8 2. 6 3. 9 4. 7 5. 12 6. 53 Problem of the Day Jan s yearly salary is $30,000, and it will be increased
More informationGRADE 7 MATH LEARNING GUIDE
GRADE 7 MATH Lesson 9: Properties of the Operations on Rational Numbers Time:.5 hours Pre-requisite Concepts: Operations on rational numbers About the Lesson: The purpose of this lesson is to use properties
More informationChapter 8 Systems of Equations and Inequalities
Chapter 8 Systems of Equations and Inequalities Mathematical Overview Learning to write and then solve a system of equations or inequalities is the foundation for solving real-world problems involving
More information1.1 evaluating expressions 2017 ink.notebook. August 18, page 7 page 8 Unit 1 Basic Equations and Inequalities. 1.1 Order of Operations.
1.1 evaluating expressions 2017 ink.notebook page 7 page 8 Unit 1 Basic Equations and Inequalities 1.1 Order of Operations page 9 page 10 Lesson Objectives Standards 1.1 Order of Operations Press the tabs
More informationStudent WebAdvisor Training Manual
Student WebAdvisor Training Manual Contents Logging into WebAdvisor..2 Registering for a Class Section..4 Paying on My Account. 9 Dropping a Class Section 12 1 Logging into WebAdvisor STEPS 1. Click the
More informationMath-3 Lesson 3-6 Analyze Rational functions The Oblique Asymptote
Math- Lesson - Analyze Rational functions The Oblique Asymptote Quiz: a What is the domain? b Where are the holes? c What is the vertical asymptote? y 4 8 8 a -, b = c = - Last time Zeroes of the numerator
More informationIdentifying Slope and y-intercept slope y = mx + b
Practice 1 Identifying m and b Identifying Slope and y-intercept slope y = mx + b y-intercept 1 1. For each of the following, identify the slope and y-intercept, OR use the slope and y-intercept to write
More informationDid you ever think that a four hundred year-old spider may be why we study linear relationships today?
Show Me: Determine if a Function is Linear M8221 Did you ever think that a four hundred year-old spider may be why we study linear relationships today? Supposedly, while lying in bed Rene Descartes noticed
More informationInstructor: Barry McQuarrie Page 1 of 6
Questions 1. Solve the system by graphing: 3x + y = 2 2x y = 3 2. Solve the system by graphing: x + 3y = 9 y = 1 3 x 2 3. Solve the system by graphing: y = 2x + 5 3y + 6x = 15 4. Solve the system algebraically,
More informationExample 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
More informationUnit 2 Day 6. Characteristics Of Quadratic, Even, And Odd Functions
Unit 2 Day 6 Characteristics Of Quadratic, Even, And Odd Functions 1 Warm Up 1.) Jenna is trying to invest money into the stock exchange. After some research, she has narrowed it down to two companies.
More informationMultiplying and Dividing Rational Expressions
Multiplying and Dividing Rational Expressions Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x 6 x 2 x 7 Factor each expression. 2. y 3 y 3 y 6 x 4 4. y 2 1 y 5 y 3 5.
More informationSetting up vendors. In this quick lesson. Quick lesson. Goal: Set up your vendors in Ajera so you can use them in your daily work.
Quick lesson Setting up vendors Goal: Set up your vendors in Ajera so you can use them in your daily work. In this quick lesson Step 1: Enter general information 2 Step 2: Enter the vendor's address 4
More informationWarm-Up Exercises. 1. If the measures of two angles of a triangle are 19º and 80º, find the measure of the third angle. ANSWER 81º
Warm-Up Exercises 1. If the measures of two angles of a triangle are 19º and 80º, find the measure of the third angle. 81º 2. Solve (x 2)180 = 1980. 13 Warm-Up Exercises 3. Find the value of x. 126 EXAMPLE
More informationQuick Start Guide Skim first for content and, then, use as necessary
Quick Start Guide Skim first for content and, then, use as necessary Big Picture: Not unlike accounting software, you need to enter certain information into Solo in order to perform COBRA administration.
More informationAdvanced Formulas and Functions in Microsoft Excel
Advanced Formulas and Functions in Microsoft Excel This document provides instructions for using some of the more complex formulas and functions in Microsoft Excel, as well as using absolute references
More information4-8 Similar Figures and Proportions. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Warm Up Find the cross products, and then tell whether the ratios are equal. 1. 16, 40 6 15 2. 3. 3 8, 18 46 8, 24 9 27 4. 28, 42 12 18 240
More information1-5 Parent Functions and Transformations
Describe the following characteristics of the graph of each parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. 1.
More informationCommon Core Algebra 2. Chapter 1: Linear Functions
Common Core Algebra 2 Chapter 1: Linear Functions 1 1.1 Parent Functions and Transformations Essential Question: What are the characteristics of some of the basic parent functions? What You Will Learn
More informationSection 1.8. Simplifying Expressions
Section 1.8 Simplifying Expressions But, first Commutative property: a + b = b + a; a * b = b * a Associative property: (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Distributive property: a * (b
More information1. Divide by using long division. (8x 3 + 6x 2 + 7) (x + 2)
Bellwork 0-7-4. Divide by using long division. (8x + 6x 2 + 7) (x + 2) Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic
More information2-4 Writing Linear Equations. Write an equation in slope-intercept form for the line described. 9. slope passes through (0, 5) SOLUTION:
Write an equation in slope-intercept form for the line described 9 slope passes through (0, 5) Substitute m = and (x, y) = (0, 5) in the equation y = mx + b Substitute m = and b = 5 in the equation y =
More informationState of California Quadrigae Global Enterprises, Inc. IFB STPD C Statewide Technology Procurement Division
Summary of Service: and Global satellite communications services. Geographic Availability: Global. Service Restrictions: identified now by or. 1 Basic Payper-Use Access Per phone charge for the Payper-Use.
More informationLesson 3: Logic and Reference Functions
Lesson 3: Logic and Reference Functions This Video Excel Educator - Looking Back Lesson 1 Excel Basics Lesson 2 Formulas and Functions Excel Educator - Looking Ahead Lesson 3 - Logic & Reference Functions
More informationStratford upon Avon School Mathematics Homework Booklet
Stratford upon Avon School Mathematics Homework Booklet Year: 7 Scheme: 1 Term: 1 Name: Show your working out here Homework Sheet 1 1: Write 7:43 pm using the 24 hour clock 11: Find the area of this shape.
More informationChapter 1 Section 1 Lesson: Solving Linear Equations
Introduction Linear equations are the simplest types of equations to solve. In a linear equation, all variables are to the first power only. All linear equations in one variable can be reduced to the form
More informationConcept Development: Functions: A function is a rule that assigns each input exactly one output. Stated another way: no x-values are repeated.
Learning Objective: we will compare the properties of linear functions in different ways. (G8M5L6) Concept Development: Functions: A function is a rule that assigns each input exactly one output. Stated
More informationSection 3.1 Variables, Expressions and Order of Operations
Section.1 Variables, Expressions and Order of Operations A variable is a symbol, usually a letter of the alphabet that represents some varying or undetermined number. An algebraic expression is a mathematical
More informationAdding Integers. Unit 1 Lesson 6
Unit 1 Lesson 6 Students will be able to: Add integers using rules and number line Key Vocabulary: An integer Number line Rules for Adding Integers There are two rules that you must follow when adding
More informationSection 1.5. Finding Linear Equations
Section 1.5 Finding Linear Equations Using Slope and a Point to Find an Equation of a Line Example Find an equation of a line that has slope m = 3 and contains the point (2, 5). Solution Substitute m =
More informationLesson 2.2 Exercises, pages
Lesson. Exercises, pages 100 105. Write each mixed radical as an entire radical. a) 6 5 b) 6 # 5 # 180 7 # 108 c) - 5 () # d) 5 5 # 5 8 # 5 65 # 0 150. Write each entire radical as a mixed radical, if
More informationLesson 2b Functions and Function Operations
As we continue to work with more complex functions it is important that we are comfortable with Function Notation, opertions on Functions and opertions involving more than one function. In this lesson,
More informationS.W.B.A.T: Identify the independent and dependent variable in sentence. Write a function rule for a table and a situation.
Lesson 31 Date: Mr. Jones S.W.B.A.T: Identify the independent and dependent variable in sentence. Write a function rule for a table and a situation. DO NOW 1. If ( ), find f(3). 2. If f(x) = 2x -1, what
More informationClick to edit Master title style. Click to edit Master title style
Click to edit Master title style Click to edit Master title style Active Network Management Getting Connected in the East Riding David van Kesteren Northern Powergrid Click Introduction Click to to edit
More informationKey Terms. Writing Algebraic Expressions. Example
Chapter 6 Summary Key Terms variable (6.1) algebraic expression (6.1) evaluate an algebraic expression (6.1) Distributive Property of Multiplication over Addition (6.2) Distributive Property of Multiplication
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3: Systems of Equations Mrs. Leahy 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system of equations
More informationMinnesota West Community and Technical College A GUIDE TO APPROVING, ORDERING, AND USING CELLULARAND OTHER MOBILE COMPUTING DEVICES AND SERVICES
Minnesota West Community and Technical College A GUIDE TO APPROVING, ORDERING, AND USING CELLULARAND OTHER MOBILE COMPUTING DEVICES AND SERVICES A. FAQs - Employees 1. I think I need a business cell phone.
More informationOnline Registration FAQs
Online Registration FAQs o How do we register individuals from the back end? (Over the phone) o How do we send emails to individuals on the roster? o How do we view and print a roster of individuals attending
More informationTaking Apart Numbers and Shapes
Taking Apart Numbers and Shapes Writing Equivalent Expressions Using the Distributive Property 1 WARM UP Calculate the area of each rectangle. Show your work. 1. 6 in. 2. 15 in. 12 yd 9 yd LEARNING GOALS
More information1-8 Exploring Transformations
1-8 Exploring Transformations Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Plot each point. D 1. A(0,0) 2. B(5,0) 3. C( 5,0) 4. D(0,5) 5. E(0, 5) 6. F( 5, 5) C A F E B Objectives Apply transformations
More informationGENERATING EQUIVALENT EXPRESSIONS ENGAGE NY- LESSONS 1 & 2
GENERATING EQUIVALENT EXPRESSIONS ENGAGE NY- LESSONS 1 & 2 FIND THE PRODUCT: -9.56 x 4.2 x -2 x CALCULATE: 3 6 7 + 5 1 2 CORE ENRICHMENT Interim Analysis Paper Pencil The cost (c) to hire a dog trainer
More informationAnswers for Lesson 4-1, pp
Answers for Lesson -, pp. 0 0. yes. no. yes. yes 5. no 6. yes 7. no 8. yes Exercises 9. a. no b. no c. yes 0. a. yes b. no c. yes. a. no b. no c. no. a. yes b. no c. no. a. no b. yes c. no. a. no b. no
More informationNotes Lesson 3 4. Positive. Coordinate. lines in the plane can be written in standard form. Horizontal
A, B, C are Notes Lesson 3 4 Standard Form of an Equation: Integers Ax + By = C Sometimes it is preferred that A is Positive All lines in the plane can be written in standard form. Oblique Coordinate Horizontal
More informationProcedures to become a Public Service Training Instructor
Procedures to become a Public Service Training Instructor The procedures to be a Public Service Training Instructor are defined by Policy 5202 (Legislative Rule 126CSR136) and other policies of the West
More informationMeasures of Dispersion
Lesson 7.6 Objectives Find the variance of a set of data. Calculate standard deviation for a set of data. Read data from a normal curve. Estimate the area under a curve. Variance Measures of Dispersion
More informationAlgebraically Speaking Chalkdust Algebra 1 Fall Semester
Algebraically Speaking Chalkdust Algebra 1 Fall Semester Homework Assignments: Chapter 1 The Real Number System: Lesson 1.1 - Real Numbers: Order and Absolute Value Do the following problems: # 1 9 Odd,
More informationMultiplying and Dividing Rational Expressions
Page 1 of 14 Multiplying and Dividing Rational Expressions Attendance Problems. Simplify each expression. Assume all variables are nonzero. x 6 y 2 1. x 5 x 2 2. y 3 y 3 3. 4. x 2 y 5 Factor each expression.
More information2. Find the next term in the sequence given above.
1. Find the rule for the following sequence. 2, 6, 18,... Warm Up 2. Find the next term in the sequence given above. 3. Find the 1st three terms of the sequence given by the rule: start at 3 and multiply
More informationObjectives. Vocabulary. 1-1 Exploring Transformations
Warm Up Plot each point. D Warm Up Lesson Presentation Lesson Quiz 1. A(0,0) 2. B(5,0) 3. C( 5,0) 4. D(0,5) C A B 5. E(0, 5) 6. F( 5, 5) F E Algebra 2 Objectives Apply transformations to points and sets
More informationUnit 3, Lesson 1.3 Special Angles in the Unit Circle
Unit, Lesson Special Angles in the Unit Circle Special angles exist within the unit circle For these special angles, it is possible to calculate the exact coordinates for the point where the terminal side
More information1-1. Variables and Expressions
exploration Catherine s dance team is planning a spring trip to the coast. Catherine is saving money in a bank account to pay for the trip. Her parents started her account with $100. She sells Christmas
More information2-1 Ordered Pairs. Lesson Presentation
Lesson Presentation Practice Problems Solve. A. x 8 = 19 B. 5 = a 2 C. 7 + n = 24 D. 3c 7 = 32 E. 17y + 7 = 58 x = 27 a = 7 n = 17 c = 13 y = 3 Practice Problem F A moving van travels 50 miles per hour.
More informationMicrosoft Excel Tutorial for Chapter 4 TIE-into Practice Exercises
Integrating Educational Technology into Teaching (4 th Edition) M. D. Roblyer University of Maryland University College Microsoft Excel Tutorial for Chapter 4 TIE-into Practice Exercises Created by William
More informationIn Google Sheets simple formulas can help you calculate important data. Learn how to create simple formulas in Google Sheets.
Google Sheets Creating Simple Formulas In Google Sheets simple formulas can help you calculate important data. Learn how to create simple formulas in Google Sheets. Introduction When working with numerical
More informationUnder the GDPR, you have the following rights, which we will always work to uphold:
1. INFORMATION ABOUT US Registered address: Branded Restaurants, Unit Upper 14, Mermaid Quay, Cardiff Bay, CF10 5BZ. Postal Address: Branded Restaurants, Unit Upper 14, Mermaid Quay, Cardiff Bay, CF10
More informationFamily of Functions Lesson
Family of Functions Lesson Introduction: Show pictures of family members to illustrate that even though family members are different (in most cases) they have very similar characteristics (DNA). Today
More information2.) = 7.) Find the unit rate of 6 miles in 20 minutes. 4.) 6 8 = 8.) Put in simplified exponential form (8 3 )(8 6 )
Warm Up Do you remember how to... 1.) 3 + 9 = Wobble Chairs: Braden, Weston, & Avalon 6.) Put 3,400,000 in scientific notation? 2.) 2 + 8 = 7.) Find the unit rate of 6 miles in 20 minutes. 3.) 2 17 = 4.)
More information