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MTH-4107-1 C1-C4 StraightLines II_Layout 1 10-02-03 10:22 Page 1 S MTH-4107-1 traight Lines II MTH-4107-1 STRAIGHT LINES II Mathematics Coordinator: Jean-Paul Groleau Author: Nicole Perreault Content revision: Jean-Paul Groleau Suzie Asselin Daniel Gélineau Louise Allard Updates: Jean-Paul Groleau Photocomposition and layout: Multitexte Plus Desktop publishing for updated version: P.P.I. inc. Cover page: Daniel Rémy English version: Services à la commuauté anglophone Direction de la production en langue anglaise – Translation: William Gore – Revision: Lana Georgieff Translation of updated sections: Claudia de Fulviis Reprint: 2006 © Société de formation à distance des commissions scolaires du Québec All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction, by mechanical or electronic means, including microreproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commissions scolaires du Québec (SOFAD). Legal Deposit — 2006 Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada ISBN 2-89493-285-8 MTH-4107-1 Answer Key Straight Lines II TABLE OF CONTENTS Introduction to the Program Flowchart ................................................... 0.4 The Program Flowchart ............................................................................ 0.5 How to Use This Guide ............................................................................. 0.6 General Introduction ................................................................................. 0.9 Intermediate and Terminal Objectives of the Module ............................ 0.11 Diagnostic Test on the Prerequisites ....................................................... 0.13 Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.19 Analysis of the Diagnostic Test Results ................................................... 0.23 Information for Distance Education Students ......................................... 0.25 UNITS 1. 2. 3. 4. Determining the Equation of a Line ........................................................ 1.1 Perpendicular and Parallel Lines............................................................. 2.1 Distance Between Two Points .................................................................. 3.1 Coordinates of a Point That Divides a Line Segment in a Particular Ratio ......................................................................................... 4.1 5. Solving Analytical Geometry Problems ................................................... 5.1 Final Review .............................................................................................. 6.1 Answer Key for the Final .......................................................................... 6.8 Terminal Objectives .................................................................................. 6.12 Self-Evaluation Test.................................................................................. 6.15 Answer Key for the Self-Evaluation Test ................................................ 6.25 Analysis of the Self-Evaluation Test Results .......................................... 6.31 Final Evaluation........................................................................................ 6.32 Answer Key for the Exercises ................................................................... 6.33 Glossary ..................................................................................................... 6.81 List of Symbols .......................................................................................... 6.86 Bibliography .............................................................................................. 6.87 Review Activities ....................................................................................... 7.1 © SOFAD 0.3 1 2 MTH-4107-1 3 Answer Key Straight Lines II INTRODUCTION TO THE PROGRAM FLOWCHART Welcome to the World of Mathematics! This mathematics program has been developed for the adult students of the Adult Education Services of school boards and distance education. The learning activities have been designed for individualized learning. If you encounter difficulties, do not hesitate to consult your teacher or to telephone the resource person assigned to you. The following flowchart shows where this module fits into the overall program. It allows you to see how far you have progressed and how much you still have to do to achieve your vocational goal. There are several possible paths you can take, depending on your chosen goal. The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2 (MTH-416), and leads to a Diploma of Vocational Studies (DVS). The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2 (MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School Diploma (SSD), which allows you to enroll in certain Cegep-level programs that do not call for a knowledge of advanced mathematics. The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2 (MTH-536), and leads to Cegep programs that call for a solid knowledge of mathematics in addition to other abilities. If this is your first contact with this mathematics program, consult the flowchart on the next page and then read the section “How to Use This Guide.” Otherwise, go directly to the section entitled “General Introduction.” Enjoy your work! 0.4 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II THE PROGRAM FLOWCHART CEGEP MTH-5112-1 MTH-5111-2 MTH-536 MTH-5104-1 MTH-5103-1 Introduction to Vectors MTH-5109-1 Geometry IV MTH-5108-1 Trigonometric Functions and Equations MTH-5107-1 Exponential and Logarithmic Functions and Equations Optimization II MTH-5106-1 Real Functions and Equations Probability II MTH-5105-1 Conics MTH-5102-1 Statistics III MTH-5101-1 MTH-436 MTH-426 MTH-4110-1 MTH-216 MTH-116 © SOFAD The Four Operations on Algebraic Fractions MTH-4109-1 Sets, Relations and Functions Quadratic Functions MTH-4107-1 Straight Lines II MTH-4106-1 Factoring and Algebraic Functions MTH-4105-1 Exponents and Radicals MTH-4103-1 MTH-4102-1 MTH-4101-2 Complement and Synthesis I MTH-4108-1 MTH-4104-2 MTH-314 Optimization I MTH-4111-2 Trades DVS MTH-416 Complement and Synthesis II MTH-5110-1 MTH-526 MTH-514 Logic You ar e h er e Statistics II Trigonometry I Geometry III Equations and Inequalities II MTH-3003-2 Straight Lines I MTH-3002-2 Geometry II MTH-3001-2 The Four Operations on Polynomials MTH-2008-2 Statistics and Probabilities I MTH-2007-2 Geometry I MTH-2006-2 Equations and Inequalities I MTH-1007-2 Decimals and Percent MTH-1006-2 The Four Operations on Fractions MTH-1005-2 The Four Operations on Integers 0.5 25 hours = 1 credit 50 hours = 2 credits 1 Answer Key 2 MTH-4107-1 3 Straight Lines II HOW TO USE THIS GUIDE Hi! My name is Monica and I have been asked to tell you about this math module. What’s your name? Whether you are registered at an adult education center or at Formation à distance, ... Now, the module you have in your hand is divided into three sections. The first section is... I’m Andy. ... you have probably taken a placement test which tells you exactly which module you should start with. ... the entry activity, which contains the test on the prerequisites. 0.6 You’ll see that with this method, math is a real breeze! My results on the test indicate that I should begin with this module. By carefully correcting this test using the corresponding answer key, and recording your results on the analysis sheet ... © SOFAD 1 Answer Key 2 MTH-4107-1 3 ... you can tell if you’re well enough prepared to do all the activities in the module. And if I’m not, if I need a little review before moving on, what happens then? Straight Lines II In that case, before you start the activities in the module, the results analysis chart refers you to a review activity near the end of the module. I see! In this way, I can be sure I have all the prerequisites for starting. START The starting line shows where the learning activities begin. Exactly! The second section contains the learning activities. It’s the main part of the module. ? The little white question mark indicates the questions for which answers are given in the text. The target precedes the objective to be met. The memo pad signals a brief reminder of concepts which you have already studied. ? Look closely at the box to the right. It explains the symbols used to identify the various activities. The boldface question mark indicates practice exercises which allow you to try out what you have just learned. The calculator symbol reminds you that you will need to use your calculator. ? The sheaf of wheat indicates a review designed to reinforce what you have just learned. A row of sheaves near the end of the module indicates the final review, which helps you to interrelate all the learning activities in the module. FINISH Lastly, the finish line indicates that it is time to go on to the self-evaluation test to verify how well you have understood the learning activities. © SOFAD 0.7 1 2 MTH-4107-1 3 There are also many fun things in this module. For example, when you see the drawing of a sage, it introduces a “Did you know that...” It’s the same for the “math whiz” pages, which are designed especially for those who love math. For example. words in boldface italics appear in the glossary at the end of the module... Answer Key A “Did you know that...”? Yes, for example, short tidbits on the history of mathematics and fun puzzles. They are interesting and relieve tension at the same time. Straight Lines II Must I memorize what the sage says? No, it’s not part of the learning activity. It’s just there to give you a breather. They are so stimulating that even if you don’t have to do them, you’ll still want to. And the whole module has been arranged to make learning easier. ... statements in boxes are important points to remember, like definitions, formulas and rules. I’m telling you, the format makes everything much easier. The third section contains the final review, which interrelates the different parts of the module. Great! There is also a self-evaluation test and answer key. They tell you if you’re ready for the final evaluation. Thanks, Monica, you’ve been a big help. I’m glad! Now, I’ve got to run. See you! 0.8 Later ... This is great! I never thought that I would like mathematics as much as this! © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II GENERAL INTRODUCTION STRAIGHT LINES II: ONE STEP TOWARDS ANALYTICAL GEOMETRY This module deals with straight lines. The Cartesian plane, coordinates, the abscissa, the origin, the slope and the equation of a line are concepts emphasized throughout this learning guide. You should normally be familiar with this subject matter, since it was the focus of a module that precedes this course. In this module, you will expand your knowledge of straight lines. This second course on straight lines is highly relevant, since the concepts you will learn can be applied in fields such as physics, chemistry, economics and business management. In fact, graphs can be used to represent a wide variety of situations, and the coordinates of aligned or non-aligned points in a Cartesian plane provide important information that makes it possible to interpret a given phenomenon. We will begin by studying different ways of finding the equation of a line when we are given only two pieces of information. We can determine the equation of a line given its slope and y-intercept, given its slope and a point on that line, or given two points on that line. We can also determine the equation of a line given a point on that line and the equation of another line parallel or perpendicular to it. How do we do this? You will have no trouble answering this question after reading the first two units. We will then examine the concept of distance and learn how to find the distance between two given points. The Pythagorean theorem will once again prove to be essential in this case. © SOFAD 0.9 1 2 MTH-4107-1 3 Answer Key Straight Lines II We will also examine two new mathematical formulas that involve using the coordinates of the endpoints of a line segment to determine the point that divides that segment in a particular ratio. We will first look at the formula for finding the point that divides a line segment in any given ratio and then use that formula to derive the formula for locating the midpoint of a segment. These concepts are extremely useful for finding the equation of the perpendicular bisector, median and altitude of a triangle in analytical geometry. Analytical geometry is the study of geometric figures in which algebraic reasoning is used and position is represented by coordinates. In the last unit, you will be able to put what you have learned to the test by solving problems involving several of the concepts you covered in the previous units Without further ado, let’s enter the world of analytical geometry in which you will discover a variety of new concepts. 0.10 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II INTERMEDIATE AND TERMINAL OBJECTIVES OF THE MODULE Module MTH-4107-1 contains five units and requires 25 hours of study distributed as shown below. Each unit covers either an intermediate or a terminal objective. The terminal objectives appear in boldface. Objectives Number of Hours* % (Evaluation) 1 and 2 10 40% 3 3 15% 4 5 20% 5 6 25% * One hour is allotted for the final evaluation. 1. Determining the Equation of a Line Determine the equation of a line, given any one of the following: • the slope and the y-intercept of that line • the slope and a point on that line • two points on that line 2. Perpendicular and Parallel Lines Determine the equation of a line, given either one of the following: • a point on that line and the equation of a line parallel to it, • a point on that line and the equation of a line perpendicular to it. The coefficients of the variables of these linear equations and the coordinates of the points are rational numbers. The resulting equation must be of the form y = mx + b or the form Ax + By + C = 0. The steps in the solution must be shown. © SOFAD 0.11 1 2 MTH-4107-1 3 Answer Key Straight Lines II 3. Distance Between Two Points Determine the distance between two given points in the Cartesian plane. The coordinates of these points and the distance are rational numbers and the problems deal with everyday situations. The steps in the solution must be shown and the distance must be stated in a unit of measure. 4. Coordinates of a Point That Divides a Line Segment in a Particular Ratio Determine the point that divides a line segment in a particular ratio, given the endpoints of the line segment. The coordinates of the endpoints and the ratio are rational numbers and the problems deal with everyday situations. The ratio in which the segment is divided must be derived from the information given in the problem. The steps in the solution must be shown. 5. Solving Analytical Geometry Problems Solve problems that involve calculating the distance between two points, determining the coordinates of a point that divides a line segment and finding the equation of a line. The solutions may involve all or some of these concepts. 0.12 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II DIAGNOSTIC TEST ON THE PREREQUISITES Instructions 1. Answer as many questions as you can. 2. You may use a calculator. 3. Write your answers on the test paper. 4. Don't waste any time. If you cannot answer a question, go on to the next one immediately. 5. When you have answered as many questions as you can, correct your answers using the answer key which follows the diagnostic test. 6. To be considered correct, your answers must be identical to those in the key. In addition, the various steps in your solution should be equivalent to those shown in the answer key. 7. Transcribe your results onto the chart which follows the answer key. This chart gives an analysis of the diagnostic test results. 8. Do only the review activities that apply to each of your incorrect answers. 9. If all your answers are correct, you may begin working on this module. © SOFAD 0.13 1 2 MTH-4107-1 3 Answer Key Straight Lines II 1. Plot the following points in the Cartesian plane below. y A(2, 4) B 3, – 5 2 C(–2, 0) 1 1 x D(0, 0) E – 1 , –4 2 2. Graph the following equations in the Cartesian plane below. a) 3y + 9x = – 12 b) y + 6 = – 3x ................................................... ........................................................ ................................................... ........................................................ x x y y 0.14 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II y 1 1 x c) What kind of lines did you draw in a) and b)? Explain your answer. ....................................................................................................................... ....................................................................................................................... 3. Find the slope of each of the following lines. a) 2x – 3y – 2 = 0 ....................................................................................................................... ....................................................................................................................... b) y = – x + 13 4 ....................................................................................................................... ....................................................................................................................... © SOFAD 0.15 1 2 MTH-4107-1 3 Answer Key Straight Lines II 4. What is the slope of the line that passes through each of the following points? a) (4, – 2) and (– 1, 0) ....................................................................................................................... ....................................................................................................................... b) 1, 1 2 3 and 0, 2 3 ....................................................................................................................... ....................................................................................................................... 5. Solve the following equations, showing all the steps in your solution. a) y–2 = 3 2y + 4 2 b) 3x – 4 = 8 – 2x 5 3 ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ 0.16 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II 6. Calculate the length of the third side of the right triangles below. a) b) C 2 A C ? 5 ? 9 A 6 B B ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ ................................................... ........................................................ 7. The Fords are going away for the weekend. They will be travelling two hundred and fifty kilometres due south and three hundred and eighty kilometres east. They can cover four hundred and seventy-five kilometres on one tank of gas and it costs them thirty-two dollars to fill up. Calculate the straight-line distance (d) between their home and their destination. ................................................................ ................................................................ ................................................................ ................................................................ ................................................................ ........................................................................................................................... © SOFAD 0.17 1 2 3 Answer Key 1 Answer Key 2 MTH-4107-1 3 Straight Lines II ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES y 1. • C(–2, 0) 1 • • A(2, 4) D(0, 0) 1 x 3, – 5 2 •B • E – 1, –4 2 2. a) 3y + 9x = – 12 b) y + 6 = – 3x 3y = – 9x – 12 y = – 3x – 6 y = – 3x – 4 x –2 –1 0 x –2 –1 0 y 2 –1 –4 y 0 –3 –6 y b) a) • 1 • 1 • • • • © SOFAD 0.19 x 1 2 MTH-4107-1 3 Answer Key Straight Lines II c) They are parallel, distinct lines because they have the same slope (m = – 3) and different y-intercepts (– 4 ≠ – 6). 3. a) m = – A = – 2 = 2 (Equation of the form Ax + By + C = 0) –3 3 B b) m = coefficient of x = – 1 (Equation of the form y = mx + b) 4 y –y 0 – (– 2) 4. a) m = x2 – x1 = = 2 = –2 2 1 –1 – 4 –5 5 2– 1 1 y2 – y1 3 3 = 3 = 1 × –2 = – 2 b) m = x – x = 2 1 1 3 1 3 0– –1 2 2 5. a) y–2 = 3 2 2y + 4 b) 3x – 4 = 8 – 2x 5 3 2(y – 2) = 3(2y + 4) 3(3x – 4) = 5(8 – 2x) 2y – 4 = 6y + 12 9x – 12 = 40 – 10x 2y – 6y = 12 + 4 9x + 10x = 40 + 12 – 4y = 16 y = 16 –4 y = –4 19x = 52 x = 52 or 2 14 19 19 6. a) a2 = b2 + c2 b) c2 = a2 – b2 a2 = 52 + 62 c2 = 92 – 22 a2 = 25 + 36 c2 = 81 – 4 a2 = 61 c2 = 77 a = 7.81 c = 8.77 0.20 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II N 7. Home W E d2 = 2502 + 3802 d 250 380 S d2 = 62 500 + 144 400 Destination d2 = 206 900 d = 454.86 (to the nearest hundredth) Their home and destination are 454.9 kilometres apart. © SOFAD 0.21 1 2 3 Answer Key 1 2 MTH-4107-1 3 Answer Key Straight Lines II ANALYSIS OF THE DIAGNOSTIC TEST RESULTS Answers Questions Correct Incorrect 1. 2. a) b) c) 3. a) b) 4. a) b) 5. a) b) 6. a) b) 7. Review Section 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.3 7.3 7.1 7.1 7.1 Page 7.19 7.19 7.19 7.19 7.19 7.19 7.19 7.19 7.32 7.32 7.4 7.4 7.4 Before going on to unit Unit 1 Unit 1 Unit 1 Unit 1 Unit 1 Unit 1 Unit 1 Unit 1 Unit 1 Unit 1 Unit 3 Unit 3 Unit 3 • If all your answers are correct, you may begin working on this module. • For each incorrect answer, find the related section listed in the “Review” column. Complete this section before beginning the unit listed in the righthand column under the heading “Before Going on to Unit”. © SOFAD 0.23 1 2 3 Answer Key 1 2 MTH-4107-1 3 Answer Key Straight Lines II INFORMATION FOR EDUCATION STUDENTS DISTANCE You now have the learning material for MTH-4107-1 together with the homework assignments. Enclosed with this material is a letter of introduction from your tutor indicating the various ways in which you can communicate with him or her (e.g. by letter or telephone) as well as the times when he or she is available. Your tutor will correct your work and help you with your studies. Do not hesitate to make use of his or her services if you have any questions. DEVELOPING EFFECTIVE STUDY HABITS Distance education is a process which offers considerable flexibility, but which also requires active involvement on your part. It demands regular study and sustained effort. Efficient study habits will simplify your task. To ensure effective and continuous progress in your studies, it is strongly recommended that you: • draw up a study timetable that takes your working habits into account and is compatible with your leisure time and other activities; • develop a habit of regular and concentrated study. © SOFAD 0.25 1 2 MTH-4107-1 3 Answer Key Straight Lines II The following guidelines concerning the theory, examples, exercises and assignments are designed to help you succeed in this mathematics course. Theory To make sure you thoroughly grasp the theoretical concepts: 1. Read the lesson carefully and underline the important points. 2. Memorize the definitions, formulas and procedures used to solve a given problem, since this will make the lesson much easier to understand. 3. At the end of an assignment, make a note of any points that you do not understand. Your tutor will then be able to give you pertinent explanations. 4. Try to continue studying even if you run into a particular problem. However, if a major difficulty hinders your learning, ask for explanations before sending in your assignment. Contact your tutor, using the procedure outlined in his or her letter of introduction. Examples The examples given throughout the course are an application of the theory you are studying. They illustrate the steps involved in doing the exercises. Carefully study the solutions given in the examples and redo them yourself before starting the exercises. 0.26 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II Exercises The exercises in each unit are generally modelled on the examples provided. Here are a few suggestions to help you complete these exercises. 1. Write up your solutions, using the examples in the unit as models. It is important not to refer to the answer key found on the coloured pages at the end of the module until you have completed the exercises. 2. Compare your solutions with those in the answer key only after having done all the exercises. Careful! Examine the steps in your solution carefully even if your answers are correct. 3. If you find a mistake in your answer or your solution, review the concepts that you did not understand, as well as the pertinent examples. Then, redo the exercise. 4. Make sure you have successfully completed all the exercises in a unit before moving on to the next one. Homework Assignments Module MTH-4107-1 contains three assignments. The first page of each assignment indicates the units to which the questions refer. The assignments are designed to evaluate how well you have understood the material studied. They also provide a means of communicating with your tutor. When you have understood the material and have successfully done the pertinent exercises, do the corresponding assignment immediately. Here are a few suggestions. 1. Do a rough draft first and then, if necessary, revise your solutions before submitting a clean copy of your answer. © SOFAD 0.27 1 2 MTH-4107-1 3 Answer Key Straight Lines II 2. Copy out your final answers or solutions in the blank spaces of the document to be sent to your tutor. It is preferable to use a pencil. 3. Include a clear and detailed solution with the answer if the problem involves several steps. 4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you. In the section “Student’s Questions”, write any questions which you may wish to have answered by your tutor. He or she will give you advice and guide you in your studies, if necessary. In this course Homework Assignment 1 is based on units 1 and 2. Homework Assignment 2 is based on units 3 to 5. Homework Assignment 3 is based on units 1 to 5. CERTIFICATION When you have completed all the work, and provided you have maintained an average of at least 60%, you will be eligible to write the examination for this course. 0.28 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II START UNIT 1 DETERMINING THE EQUATION OF A LINE 1.1 SETTING THE CONTEXT Temperature Scales and Linear Equations Steve is starting to feel the heat! He has to hand in his math homework tomorrow and doesn’t really know how to solve the three problems he has been assigned even though his teacher, Mr. Mercurio, clearly explained the concepts required to do this work. Steve feels a bit lost in a world where terms like slope, abscissa, ordinate, and intercept points are just words on a page. The three problems are outlined on the next page. © SOFAD 1.1 °C 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 1 2 MTH-4107-1 3 Answer Key Straight Lines II 1. Find the equation of the line 3. When placed in melting ice, a that shows how temperature thermometer with a Celsius readings in degrees Kelvin scale reads 0°, while a ther- vary as a function of tem- mometer with a Fahrenheit perature readings in degrees scale reads 32°. When these Celsius. The slope of this line thermometers are immersed is 1 and its y-intercept is 273. in boiling water, they read 100°C and 212°F respec- 2. What is the equation of the tively. Find the equation of line that shows how tem- the line that shows how tem- perature readings in degrees perature readings in degrees Réaumur vary as a function Fahrenheit vary as a function of temperature readings in of temperature readings in degrees Celsius? The slope of 4 this line is 5 . When placed in boiling water, a thermom- degrees Celsius. eter with a Celsius scale reads 100°, whereas a thermometer with a Réaumur scale reads 80°. To achieve the objective of this unit, you should be able to find the equation of a line given its slope and y-intercept, given the slope and the coordinates of a point on that line, or given the coordinates of two points on that line. 1.2 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II Let’s return to Steve’s first problem. The slope and the y-intercept are given. The Cartesian plane in Figure 1.1 illustrates this situation. y °K 400 300 m=1 (0, 273) (y-intercept) 200 100 Origin 6 Fig. 1.1 12 °C 18 24 x Temperature readings in degrees Kelvin as a function of temperature readings in degrees Celsius • The y-intercept is the second coordinate (or ordinate) of the point where the line crosses the y-axis. This point is usually expressed as follows: (0, b). • The x-intercept is the first coordinate (or abscissa) of the point where the line crosses the x-axis. This point is usually expressed as follows: (a, 0). • The slope (or rate of change) of a line is the change in the ordinates (or y-coordinates) divided by the change in the abscissas (or x-coordinates). This change is represented by y –y the mathematical equation m = x2 – x1 . 2 1 We already know that there are two ways of writing the equation of a line. y = mx + b or Ax + By + C = 0 © SOFAD 1.3 1 Answer Key 2 MTH-4107-1 3 Straight Lines II In the first equation, m represents the slope of the line and b represents the y-intercept, namely the y-coordinate of the point where the line crosses the yaxis. If we are given the slope and the y-intercept of a line, as is the case in Steve’s first problem, we can easily find the equation of that line. 1. Let m = the value of the slope and b = the value of the y-intercept in the equation y = mx + b. Since m = 1 and b = 273, y = 1x + 273 or y = x + 273. 2. Check the resulting equation by replacing each variable with the corresponding coordinate of the point (0, 273). y = x + 273 273 = 0 + 273 273 = 273 N.B. All the points on the line satisfy this equation. You can check this for yourself, using any point on the line in Figure 1.1. In any equation of the form y = mx + b, m is the slope and b is the y-intercept. The slope of a line is the coefficient of x and the y-intercept is the constant term when the y variable is isolated. 1.4 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II In any equation of the form Ax + By + C = 0, known as the general form of the equation of a line, the slope is equal to – A if B – C B ≠ 0 and the y-intercept is equal to if B ≠ 0. B Example 1 Find the equation of a line that crosses the y-axis at the point (0, – 2.5) and that has a slope of – 3. If m = – 3 and b = – 2.5, then y = – 3x – 2.5. Check: – 2.5 = – 3(0) – 2.5 – 2.5 = – 2.5 • Lines rising to the right have a positive slope (m > 0). • Lines falling to the right have a negative slope (m < 0). • The slope of a horizontal line is zero (m = 0). • The slope of a vertical line is undefined. Fig. 1.2 Position of a line depending on its slope That was fairly easy! Now let’s move on to Steve’s second problem, which involves finding the equation of a line, given its slope and a point on that line. © SOFAD 1.5 1 Answer Key 2 MTH-4107-1 3 Straight Lines II The problem states that when a thermometer with a Celsius scale reads 100, the one with a Réaumur scale reads 80. This corresponds to the point (100, 80) in the Cartesian plane. The point (80, 100) is not represented in this situation, because the problem calls for the equation that shows how temperature readings in degrees Réaumur vary as a function of temperature readings in degrees Celsius. This means that the Réaumur readings are on the y-axis and the Celsius readings are on the x-axis and not vice-versa. Compare figures 1.3 and 1.4. °R °C • 80 • 100 (100, 80) 20 (80, 100) 20 20 100 °C Fig. 1.3 Temperature readings in degrees Réaumur as a function of temperature readings in degrees Celsius 20 80 °R Fig. 1.4 Temperature change in degrees Celsius in relation to temperature change in degrees Réaumur We already know that the slope is the vertical change (change in y or rise) divided by the horizontal change (change in x or run). In other words, y –y m = x2 – x1 . In this case, the slope is 4 , which means that the slope between the 2 1 5 point (100, 80) and any other point (x, y) on the line will also be 4 . Therefore, we 5 can say: 4 = y – 80 5 x – 100 1.6 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II To find the equation of the line, you need only solve this equation expressed as a proportion. Solving an equation expressed as a proportion involves applying the fundamental property of proportions, which states that the product of the extremes is equal to the product of the means. Hence, 4x = 6 7 2 4x × 7 = 2 × 6 28x = 12 x = 12 28 x= 3 7 Therefore: 4 = y – 80 5 x – 100 5(y – 80) = 4(x – 100) 5y – 400 = 4x – 400 5y = 4x – 400 + 400 5y = 4x + 0 y= 4x 5 N.B. In this case, b = 0. This means that the line passes through the origin, namely the point (0, 0). After finding the equation of the line, you have to check it by replacing each variable with the corresponding coordinate of the point given in the statement of the problem, i.e., (100, 80). y= 4x 5 80 = 4 × 100 5 80 = 80 © SOFAD 1.7 1 Answer Key 2 MTH-4107-1 3 Straight Lines II Hence, we can say that the equation represents a line that passes through the point (100, 80) and has a slope of 4 . By examining the figure below, you will see 5 that any point on the line satisfies this equation. °R • (100, 80) m = 4/5 • (0, 0) °C Fig.1.5 Temperature readings in degrees Réaumur as a function of temperature readings in degrees Celsius To find the equation of a line when its slope m and the coordinates (x1, y1) of a point on that line are known, simply use the slope formula by letting (x, y) represent y– y any point on that line. Hence, m = x – x1 . 1 Example 2 What is the equation of the line that has a slope of – 5 and that passes 6 through the point 2 , – 1 ? 3 2 y– y 1. Apply the formula m = x – x1 where (x1, y1) are the coordinates of the 1 given point. 1.8 © SOFAD 1 Answer Key 2 MTH-4107-1 3 –5 = 6 Straight Lines II y – –1 2 2 x– 3 y+ 1 5 2 – = 2 6 x– 3 = –5 x – 2 3 6 y+ 1 2 6y + 3 = – 5x + 10 3 6y = – 5x + 10 – 3 3 6y = – 5x + 1 3 y = –5x + 1 6 18 2. Check the equation by replacing each variable with the corresponding coordinate of the given point. y = –5x + 1 6 18 –1 = –5 2 + 1 2 6 3 18 – 1 = – 10 + 1 18 18 2 –1 = – 9 18 2 –1 = –1 2 2 Steve is now more confident and doesn’t find these problems so difficult after all. By doing the following exercises, you will see whether you too can find the equation of a line using certain information. Before going on to the exercise, however, let’s look at a bit of history. © SOFAD 1.9 1 Answer Key 2 MTH-4107-1 3 Straight Lines II Did you know that... ... temperature scales are almost always named after the scientist who devised them? Swedish astronomer and physicist Anders Celsius (1701-1744) came up with the idea of dividing a thermometer into 100 equal units. The Fahrenheit scale was created by German physicist Gabriel Daniel Fahrenheit (1686-1736). He perfected the thermometer by using mercury rather than alcohol and devised a new scale. He also invented a hydrometer, an instrument used to measure the specific gravity of liquids. The Kelvin scale, also known as the absolute scale, was devised and first used by William Thomson, alias Lord Kelvin (1824-1907). This British physicist made significant contributions to the study of thermodynamics and invented a galvanometer (a device that measures small electric currents) as well as an electrometer (an instrument used to measure differences in electric potential). French physicist and naturalist René Antoine Ferchault de Réaumur (1683-1757) also invented a thermometer that bears his name, and his work on steel and tin production made him “the father of metallurgy.” He also did a great deal of research on invertebrates. 1.10 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II Exercise 1.1 1. Find the equation of each line below, given its y-intercept and slope. Check your answer. a) The y-intercept is 5 and the slope is 2. b) The y-intercept is – 3 and the slope is 1 . 2 c) The y-intercept is – 5.35 and the slope is – 0.54. 2. What is the equation of the line that satisfies each set of conditions stated below? Check your answer. a) The line passes through (2, 5) and m = – 1 . 2 © SOFAD 1.11 1 2 MTH-4107-1 3 Answer Key Straight Lines II b) The line passes through (– 2, –1) and m = 2. c) The line passes through 1 , 1 3 4 and m = – 1 . 6 d) The line passes through (0, 3) and m = – 8. e) The line passes through (– 2.4, 6.3) and m = 5.6. 1.12 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II Steve is now set to tackle the third problem in his homework assignment, which involves finding the equation of the line that shows how temperature readings in degrees Fahrenheit vary as a function of temperature readings in degrees Celsius. ? Along which axis of the Cartesian plane should the Fahrenheit readings be scaled? ........................................................................................................................... ? Along which axis of the Cartesian plane will the Celsius readings be scaled? ........................................................................................................................... Since the problem states that the line should show how temperature readings in degrees Fahrenheit vary as a function of temperature readings in degrees Celsius, the Fahrenheit readings should be scaled along the y-axis and the Celsius readings along the x-axis. The problem also states that when placed in melting ice, the Fahrenheit thermometer reads 32° and the Celsius thermometer reads 0°. This means that one of the points on the graph is (0, 32). In boiling water, these thermometers read 212 degrees Fahrenheit and 100 degrees Celsius respectively. These measurements correspond to the point (100, 212). Steve seems to remember Mr. Mercurio saying that they should draw the line representing a given situation because a diagram often makes it easier to solve the problem. ? Plot the points (0, 32) and (100, 212) in the following Cartesian plane and draw the line that passes through these two points. © SOFAD 1.13 1 Answer Key 2 MTH-4107-1 3 Straight Lines II °F °C Fig. 1.6 Cartesian plane We need to find the equation of the line that passes through these two points. The figure below shows the line representing the situation described in the problem. °F 220 200 180 160 140 120 100 80 60 40 • (0, 32) 20 • (100, 212) 10 20 30 40 50 60 70 80 90 100 Fig. 1.7 °C Temperature readings in degrees Fahrenheit as a function of temperature readings in degrees Celsius 1.14 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II Find the equation of a line, given two points on that line. 1. Calculate the slope of this line, using the coordinates of the two points. (x1, y1) = (0, 32) and (x2, y2) = (100, 212). y –y m = x2 – x1 2 1 m = 212 – 32 100 – 0 m = 180 100 m= 9 5 N.B. For the sake of convenience, the points (0, 32) and (100, 212) have been designated as (x1, y1) and (x2, y2) respectively so that the difference between the y-coordinates and the difference between the x-coordinates would be positive. Nevertheless, we would get the same result if we designated the points (100, 212) and (0, 32) as (x1, y1) and (x2, y2) respectively. 2. Use the value of the slope and the coordinates of one of the two points to find the equation. y–y m = x – x1 1 9 = y – 32 5 x–0 5(y – 32) = 9 (x – 0) 5y – 160 = 9x 5y = 9x + 160 y = 9 x + 160 5 5 y = 9 x + 32 5 © SOFAD 1.15 1 Answer Key 2 MTH-4107-1 3 Straight Lines II N.B. We can also use the point (100, 212) to find the equation. The calculations are as follows: 9 = y – 212 5 x – 100 5(y – 212) = 9(x – 100) 5y – 1 060 = 9x – 900 5y = 9x + 160 y = 9 x + 160 5 5 9 y = x + 32 5 N.B. Because (0, 32) is the y-intercept, we can also find the equation of the line by letting m = 9 and b = 32. Hence, 5 y = mx + b y = 9 x + 32 5 3. Check the resulting equation by replacing the variables with the coordinates of either of the given points. y = 9 x + 32 5 32 = 9 (0) + 32 5 32 = 32 N.B. Of course, you can check your answer using the point (100, 212). Example 3 What is the equation of the line that passes through the points (2.53, – 3.02) and (2.45, 0)? 1.16 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II 1. Calculate the slope of the line passing through these two points. y –y m = x2 – x1 2 1 m= 0 – (– 3.02) 2.45 – 2.53 m = 3.02 – 0.08 m = – 37.75 2. Use the value of the slope and the coordinates of one of the points to find the equation. y–y m = x – x1 1 – 37.75 = y–0 x – 2.45 y = – 37.75(x – 2.45) y = – 37.75x + 92.49 N.B. We get 92.49 by rounding off to the nearest hundredth. 3. Check the resulting equation by replacing the variables with the coordinates of either of the given points. y = – 37.75x + 92.49 0 = – 37.75(2.45) + 92.49 0 = – 92.49 + 92.49 0=0 © SOFAD 1.17 1 2 MTH-4107-1 3 Answer Key Straight Lines II Steve has finished his homework, realizing that there was no great mystery to this after all! Once you understand the terms related to Cartesian planes, all you have to do is solve equations expressed as proportions. Now let’s do a few exercises to see if you can manage as well as Steve did. Exercise 1.2 Determine the equation of the line that passes through each pair of points given below. Show all the steps in your solution and check your answer. N.B. Your results must be rounded to the nearest hundredth, when necessary. 1. (1, – 3) and (2, 2) 2. (2.3, – 5,1) and (3.2, 2.7) 1.18 © SOFAD 1 2 MTH-4107-1 3. 3 Answer Key 1 , – 1 and – 3 , 2 2 2 4. (0, 4) and 1 , 5 2 2 5. (2.1, – 3.4) and (– 1.3, – 2.7) © SOFAD 1.19 Straight Lines II 1 2 MTH-4107-1 3 Answer Key Straight Lines II 6. (0, 3.25) and (5.14, 0) 7. 0, 1 3 and 1, 0 8 8. (0, 0) and (– 3.2, – 1.3) 1.20 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II When trying to find the equation of a line, you may come across two special situations. The following two examples illustrate these cases. Example 4 What is the equation of the line that passes through the points (–3, 3) and (4, 3)? 1. First, calculate the slope. m= 3–3 = 0 = 0 =0 4 – (– 3) 4 + 3 7 Since its slope is equal to 0, this line is horizontal. 2. Find its equation. y–3 x+3 y – 3 = 0(x + 3) 0= y–3=0 y=3 y m=0 • (–3, 3) Equations of the form y = b, namely those which do not have an x variable, are represented by a horizontal • (4, 3) 1 0 1 x line parallel to the x-axis. Fig. 1.8 Horizontal line The second part of this example is easier to do. Since the distance between the x-axis and each point on a horizontal line is the same, all the points have the same y-coordinate. If b = 3 and y = b , then y = 3. © SOFAD 1.21 1 2 MTH-4107-1 3 Answer Key Straight Lines II Example 5 What is the equation of the line that passes through the points (1, 4) and (1, – 2)? 1. First, calculate the slope. m = – 2 – 4 = – 6 , which is undefined. 0 1–1 Since its slope is undefined, this line is vertical. y Equations of the form x = a, namely • m: undefined those which do not have a y variable, are represented by a vertical line par- 1 0 allel to the y-axis. (1, 4) 1 • x (1, –2) Fig. 1.9 Vertical line 2. Find its equation. Since the distance between the y-axis and each point on a vertical line is the same, all the points have the same x-coordinate. If a = 1 and x = a, then x = 1. By simply looking at the coordinates, find the equation of the line that passes through each of the following pairs of points. ? a) (– 2, 5) and –2, 3 : ................... b) (0.25, 3.75) and (4, 3.75): .............. 4 1.22 © SOFAD 1 Answer Key 2 MTH-4107-1 3 Straight Lines II If your equations are x = – 2 and y = 3.75, you have used the quickest way of finding the solution. In the first set of ordered pairs, the x-coordinate (– 2) remains the same, whereas in the second set of points, the y-coordinate (3.75) remains the same. You don’t have to do any calculations in these situations. You need only remember that x = a represents a vertical line and that y = b corresponds to a horizontal line. Before going on to the practice exercises, let’s summarize the steps involved in finding the equation of a line when different types of information are given. Procedure for finding the equation of a line, given its slope and its y-intercept: 1. Let m = the value of the slope and b = the value of the y-intercept in the equation y = mx + b; 2. Check the resulting equation by replacing each variable with the corresponding coordinate of the point (0, b). Procedure for finding the equation of a line, given its slope and a point on that line: y– y 1. Apply the formula m = x – x1 where (x1, y1) are the 1 coordinates of the given point; 2. Check the resulting equation by replacing each variable with the corresponding coordinate of the given point. © SOFAD 1.23 1 Answer Key 2 MTH-4107-1 3 Straight Lines II Procedure for finding the equation of a line, given two points on that line: 1. Calculate the slope of the line by substituting the coordinates of the two given points for the variables in the y –y formula m = x2 – x1 ; 2 1 2. Find the equation of the line by substituting the value of the slope and the coordinates of one of the given points in y–y the formula m = x – x1 ; 1 3. Check the resulting equation by replacing the variables with the coordinates of either of the given points. We’ve now come to the end of this unit. The practice exercises will test your understanding of the concepts described so far. If you have trouble with these exercises, reread the relevant explanations. You have to understand these concepts to be able to do the rest of the work in this course. For now though, let’s take a look at the world of science. Did you know that… …American scientists have discovered a relationship between air temperature and the number of times a cricket chirps per minute? This means that you could use a cricket as a thermometer! 1.24 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II If a cricket chirps 40 times per minute at 50°F and 140 times per minute at 75°F, find the equation that American scientists devised to show how the temperature in degrees Fahrenheit is related to the number of times a cricket chirps per minute. Using this equation, determine the temperature (first in degrees Fahrenheit, then in degrees Celsius) at which a cricket stops chirping. N.B. The solution is in the answer key for the exercises. © SOFAD 1.25 1 2 MTH-4107-1 ? 1.2 3 Answer Key Straight Lines II PRACTICE EXERCISES For each of the following problems, find the equation of the line that satisfies the given conditions. Show all the steps in your solution and check your answers. N.B. The results must be rounded off to the nearest hundredth, when necessary. 1. The line has a slope of 3 and passes through the point (– 2, 7). 2. The line has a slope of – 2 and passes through the point (2, – 5). 3. The line has a slope of – 4 and passes through the point (0, 4). 3 1.26 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II 4. The line passes through the points (– 2, 6) and (3, – 5). 5. The line passes through the points (2.3, – 1.5) and (1.7, 2.8). © SOFAD 1.27 1 2 MTH-4107-1 3 Answer Key Straight Lines II 6. The line has a slope of – 2 and its y-intercept is 4. 3 7. The line has a slope of 1 and its x-intercept is – 3. 3 8. The y-intercept is – 2.5 and the x-intercept is – 5.3. 9. The line passes through the points (– 3, 2) and (4, 2). 1.28 © SOFAD 1 2 MTH-4107-1 3 Answer Key Straight Lines II 10. The y-intercept is – 1 and the line passes through the point (3, – 2). 2 11. The line has a slope of – 2 and passes through the origin. 3 12. The line has a slope of – 2.3 and passes through the point (2.5, 0.4). 13. The line passes through the points – 5 , 7 and – 5 , 0 . 2 2 © SOFAD 1.29 1 2 MTH-4107-1 3 Answer Key Straight Lines II 14. The line has a slope of 0 and passes through the point – 1 , – 2 . 2 3 15. The height of the column of mercury in a thermometer is zero at 0°C and 11 cm when the temperature rises to 33°C. Find the equation of the line for each case below. a) The height of the column of mercury is scaled along the y-axis and the temperature in °C is scaled along the x-axis. b) The temperature in °C is scaled along the y-axis and the height of the column of mercury is scaled along the x-axis. 1.30 © SOFAD 1 Answer Key 2 MTH-4107-1 3 1.3 Straight Lines II REVIEW EXERCISES 1. Complete the following sentences by writing in the correct missing terms or expressions. The slope of a line is the ....................................... of x when the y variable is isolated. The y-intercept of a line is the ................................. ............................. when the y variable is isolated. 2. Find the equation of each line, given the information in the following table. m x-intercept b P1(x1, y1) P2(x2, y2) Equation of the line 6 –3 –3 a) 4 b) (2.3, – 3.6) –3 4 –3 2 © SOFAD (4.2, 3.7) c) 2 5 d) 1 3 e) 1.31 1 Answer Key 2 MTH-4107-1 1.4 3 Straight Lines II THE MATH WHIZ PAGE Intercept Points If a and b are respectively the intercept points of any given line, then that line passes through the points (a, 0) and (0, b). We already know that the slope determined using the coordinates of two given points is equal to the slope between any point on the line and one of the two given points. This relationship is expressed as follows: y – y1 y2 – y1 x2 – x1 = x – x1 Using this equation, find the equation of a line, given its intercept points. 1.32 © SOFAD