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LINEAR SYSTEMS Teacher's Guide Linear Systems Teacher's Guide This teacher's guide is designed for use with Linear Systems, a series of six ten-minute programs. The programs are broadcast by TVOntario, the television service of the Ontario Educational Communications Authority. For broadcast dates, consult the TVOntario School Broadcasts Calendar, which is published in September. The series is avilable on videotape to educational institutions and nonprofit organizations. The Guide Writer: Ron Carr Editor: Mei-Lin Cheung The Series Producer/Director: David Chamberlain Project Officer: Peter Puusa Ordering Information To order this publication or videotapes of the series, or for additional information, please contact the following: United States Ontario TVOntario U.S. Sales Office 901 Kildaire Farm Road Building A Cary, North Carolina 27511 Phone: 800-331-9566 Fax: 919-380-0961 E-mail: [email protected] TVOntario Sales and Licensing Box 200, Station Q Toronto, Ontario M4T 2T1 (416) 484-2613 1. 2. 3. 4. 5. 6. Program The Equalizer The Intercepts Solving Graphically Methods of Elimination Substitution and Comparison The Third Dimension e Copyright 1994 by the Ontario Educational Communications Authority. All rights reserved. Printed in Canada. Y160 BPN 463601 463602 463603 463604 463605 463606 Page Introduction 1 Program 1. The Equalizer 3 Program 2. The Intercepts 5 Program 3. Salving Graphically 7 Program 4. Methods of Elimination 11 Program 5. Substitution and Comparison 13 Program G. The Third Dimension 17 I NTRODUCTION Together with numbers, equations belong to the first mathematical achievements of civilization. Equations occur in the cuneiform texts of the old Babylonians of about 3000 B.C. and in Egyptian papyri writings from about 1800 B.C. In Babylon, questions and problems regarding the fair sharing of inheritances were of utmost importance, and solutions often involved the creation of "linear equations." Before an algebraic symbolic language had been developed, equations had to be written in words. The equality sign (=) in common use today, was proposed by Robert Recorde, (1510-1588), but it took considerable time before it was generally accepted. The six-program series Linear Systems introduces a variety of methods for solving systems of two equations with two variables. The graphing approach involves finding the point of intersection of two lines, while the algebraic methods involve elimination by addition or subtraction, substitution, or comparison. Applications are also presented. Linear Systems is another series in the Concepts in Mathematics group. As in the other series, state-of-the-art computer animation techniques are used to create interesting and informative programs. PROGRAM I THE EQUALIZER Program Synopsis The program opens in a mathematics classroom as the teacher closes the door on the way out. Mathematical problems and solutions cover the "board." Two chairs slowly "morph" into two alien beings, Cy and Lenny, who describe their mission to discover what Earth humans have learned. They emphasize that "balance is the key" in much work involving mathematics. A brief history of the need for numbers and equations is presented. A very old problem is discussed and the resulting equation, is written. The concept of a linear equation with one variable is discussed and the balancing method used to solve such equations is explained. Equivalent equations are used in this process. Linear equations with two variables are discussed. An equation with two variables defines a relation or relationship between the variables, and each pair of numbers is a solution. These equations have an infinite number of solutions which are usually written as ordered pairs and are recorded in tables of values. At the end of the program, Cy and Lenny discuss the ability to graph the ordered pairs on a Cartesian plane, resulting in a straight line. Objectives Before-Viewing Activities After viewing this program and completing several of the following activities and exercises, students should be able to 1. Before viewing this program, students should review the meaning of the following: term, variable, constant, expression, power, exponent, equation • recognize an equation as being a mathematical statement relating two equalities; • solve a linear equation with one variable b) isolating the variable using operations on both sides of the equation; • understand the concept of equivalent equations; • verify the solution of a linear equation with one variable; • understand that an equation with two variables has an infinite number of solutions, which can be written as ordered pairs. 2. Simplify the following expressions: (a) 2(x + 5) - 2x - 4 (b) (x+4)(x-3)-x2 (c) (x + 2) 2 + (2x + 1)(x - 7) 3. Solve the following equations. (a)2x+3=8 (b)3b-1=2 4. Repeat (4) for the following equations. (a)4x+3=2(2x+1)+5 (b)5(2y+7)-3(4y-3)+2(y-22)=2 (c) 2(v + 2) = 4 (d) 3x+5 =x+4 (e) 5m-3 =7m+ 11 4. Verify the solution for each equation in (3) by determining if the left side is equal to the right side, by substitution. After-Viewing Activities 5. Make up two equations, each with one variable, such that one equation has no solution, and the other equation has an infinite number of solutions. 6. Solve and verify the following equation. Draw a number line and locate the solution on the line. (2x+1)2 -4(x+3) 2 =2-x 1. Simplify. (a) (2x + 3) 2 + (2x - 6)(3x + 2) - (3 - 3x)2 (b) 2x-3 _ 4x+1 - 2x+1 3 6 2 c) 2a2 - 3a2 -2a + 5a 3 4 2 2. Solve the following linear equations. 3. How many solutions does a linear equation with one variable have? Solve the following equations. How many solutions (roots) are there for each equation? Express the solutions "set-builder" form. (a)2(x+3)+3(x+5)=5(x+1)+16 (b) 3(2 - 3a) = --2(4a + 1) - a + 8 7. Solve the equation 8. Given the equation 3x + 2y = 11, let x have the value of 3 and solve for y. Repeat, with x = 2,4,-3, and 0. List the related values in a chart (table of values), and express the solutions as ordered pairs. PROGRAM 2 THE INTERCEPTS Program Synopsis Cy and Lenny continue their discussion of mathematical concepts and agree that it is too early for them to look at systems of equations. One of the characters "morphs" into a television screen and we are presented with a review of concepts discussed in the previous program. A linear equation with one variable is solved by using the accepted methods of adding and subtracting from both sides, and multiplying and dividing both sides by appropriate numbers, until the variable is isolated on one side of the equation and the solution is on the other side. We are reminded that a linear equation with two variables will have an infinite number of solutions, usually represented as ordered pairs. These solutions can be tabulated using a table of values. An equation with two variables is presented, and solutions are listed in a table of values. The Cartesian plane is introduced with appropriate terminology (x and y axes, origin, scales). The location of points (ordered pairs) on this plane is explained. The points in the table of values are plotted and joined to form a (straight) line. Using the line, the concepts of the x- and Y- intercepts are presented. Another relation is chosen and graphed using the intercepts. Returning once again to the problem of finding ordered pairs common to two equations with two variables, Lenny and Cy discuss the possibility of creating tables of values for each equation. These could then be examined to try to find a match, an ordered pair common to both lines. Objectives Before-Viewing Activities After viewing this program and completing several suggested exercises; students should be able to 1. Rearrange the following equations into the form Ax +By+C=0. list solutions for one equation with two variables in a table of values; s construct a Cartesian plane with appropriate labels and scales; locate and name positions on the plane using ordered pairs; sketch the graph of a relation defined by a linear equation with two variables; find the x- and y- intercepts of a line, and use them to sketch its graph. (a) 4x - 5y = 6 (b) 3y + 5(2x- 3) = 0 (c)4(x-3)+3(2-2y)=x+3y-4 2. Rearrange the following equations into the form y=mx+b. (a) x+2y-3=0 (b) 2y-5=3x (c) 3(1 + y) - 4(y + 1) = 9 3. List five ordered pairs for each of the following equations: (a)y=2x+3 (b) 2x + 3y = 6 (c) 3x - y - 9 = 0 4. What are the slopes of the lines in (3)? 5. Find the x- and y-intercepts of the lines defined by the following equations. (a)x+2y=2 (b) 2x - 3y = 6 4. Which form of the equation appears to be the most appropriate to find ordered pairs? 5. Construct a Cartesian plane and locate the following points. A(3, 1), B(-2, 5), C(4, 0), D(3, -4), E(-1, -6), F(1.5, 4.5) 6. Plot the ordered pairs which you found in (3a) on a Cartesian plane. Join the points. What is the shape of the figure? (c)y=4x-8 (d)3y+5x-2=0 6. On one set of axes, sketch the graphs of the lines defined by the following. (i) x + 2y = 5 (ii) 2x + y = 4 7. Repeat (6) for the following equations. (i) 3x + y = -3 After-Viewing Activities 1. Construct a table of values containing at least five ordered pairs for each of the following. (a)y=2x+3 (b) 2x + 3y = 12 2. Sketch the graphs of the relations defined by the following equations. (a)y=3x-1 (b) 3x - y = 3 (c) 3s - 4t= 6 3. Find three ordered pairs for each of the following, and sketch their graphs on one Cartesian plane. (a)x=3 (b) y = -2 (ii) 2x - y = -10 8. (a) What characteristics of the defining equation of a relation create a graph which is a (straight) line? (b) Sketch the graphs of the relations defined by the following. (i) y = x2 (ii) y = -2x2 + x PROGRAM 3 SOLVING GRAPHICALLY Program Description Our two friends, Cy and Lenny, are still looking for an ordered pair which is common to two equations. It has taken three days so far, and the end is still not in sight. They both realize that they need help. Methods used to sketch the graph of a linear relation with two variables from the previous program are reviewed. Two equations with two variables are introduced as a system of simultaneous equations. In attempting to solve a system such as this, we are trying to find one value for x and one value for y, which will make each equation true, at the same time. We are looking for an ordered pair which is common to both equations. This is potentially very difficult and, depending on the ordered pair, perhaps almost impossible to do by listing ordered pairs for each equation and attempting to discover a match visually. Instead, we can see that the ordered pair common to both lines will be at the point of intersection (where the lines cross). A second system of equations is presented and the solution is found by graphing each line, using intercepts, and determining the coordinates of the point of intersection. Two special cases are presented. If the two lines are parallel (and distinct), then there is no point of intersection, and the system has no solution. Such a system is called inconsistent. If the two equations result in two lines in exactly the same position (one line), then we have a dependent system with an infinite number of solutions. A final example is presented to illustrate the need for a better method to find the solution for a system of equations. The lines intersect at a point which is not easily identified accurately. Objectives Before-Viewing Activities After viewing this program and completing several exercises and questions, students will be able to • understand that the method of attempting to find a matching ordered pair from two lists of points on two lines is not desirable; • understand tnat the point of intersection of two lines is the ordered pair or point which is on both lines, and is therefore the solution for the two equations; • find the solution of two equations with two variables by sketching their graphs and finding the point of intersection by inspection; • realize that, if two lines cross at a point which is not easily or accurately identified, another method to solve the equations must be found. 1. Using the intercepts, sketch the graphs of the lines defined by the following equations. (a)4x+y=8 (b) 3x - 2y = 6 (c)y=5x+5 2. Create tables of values for the following pair of equations, and attempt to find the ordered pair which is common to both tables. (i)x+4y= 1 (ii) 2x + 5y = -1 3. Sketch the graphs of the relations defined by the following pair of equations on one set of axes. (i) 4x + y = 8 (ii) 3x - 2y = -5 4. Repeat (3) for the following pair of lines. (i) 3x - 2y =10 (ii) y = 2x - 6 5. Four times a number added to twice another number gives a sum of 14. (a) What are the numbers? (Is there just one answer?) (b) Is there a solution in which both numbers are negative? (c) If the first number is -5.5, what is the second number? (d) If the second number is 4, what is the first number? (b) What is the y-intercept of this line? (c) What is its slope? (d) What is the equation of the line, in slope/ y-intercept form? (Solve for y in terms of x.) (e) What is true about lines which are parallel? (f) Write the equation of the line with slope 5 and with y-intercept 3. (g) Write the equation of a line parallel to the line in (2a). How many such lines are there? (h) Sketch the graph of your line in (2g) on the same set of axes as (2a). (i) What is the solution of these two equations? 3. Estimate the solutions of the following systems of equations. (a)x+2y=3 2x-3y=1 (b)2x+5y=2 I 3x+y = 1 After-Viewing Activities 1. By sketching the graphs of the following pairs of lines, find the solution of each system. (a)4x+y=9 4. Classify the following systems of equations as being consistent, inconsistent, or dependent. (a)y=2x-3 y =3x+2 -3x + 2y = -4 (b)y=3x+6 2x-3y=-4 (b)2x+3y=6 4x+6y=4 (c) 3x + 4y = -9 x-3y=10 (c)x-3y=5 x-4y=5 2. (a) Sketch the graph of the line defined by 4x +2y=-3. (d)y=3x+4 3x-y+4=0 (e) 3x = 2y - 2 2y=2-3x 5. Write equations which represent the following situations. Represent the variables (unknowns) by x and y. (a) The sum of two numbers is 24. (b) The difference of one number and twice another number is 10. (c) Mary is three years older than Bob. (d) Five years ago, Robert was twice as old as Razwan. (e) The distance to Toronto is 50 km more than twice the distance to Montreal. (f) One airplane is travelling at a speed 300 km/h faster than another plane. (g) Three times a number is 34 less than twice another number. 6. Solve the following equations, by eliminating the fractions as a first step. PROGRAM 4 METHODS OF ELIMINATION Program Synopsis In a continuing attempt to use the graphing method to solve a system of linear equations with two variables, Cy and Lenny have created a coordinate system which includes numbers from -1000 to 1000, with 100 lines between each number. The grid covers the walls and ceiling of a massive room. The program continues with a review of previously learned material. The attempt to find an ordered pair which is common to two linear relations would be a long search if one were to examine ordered pairs trying to find a match. In fact, depending on the solution, the match may never be found. A second method involving finding the point of intersection of the linear graphs of each relation is a reasonable method, but only if the lines intersect at a point which is easily identified. Because of these problems, an algebraic approach is discussed. (An algebraic method will produce the accurate roots, no matter how difficult it might be to find them graphically.) The method is called elimination (by addition or subtraction). Two equations are presented, and it is found that, by adding the two equations, one of the variables is eliminated. The result is an equation with just one variable which is easily solved. This value is then substituted into one of the original equations to find the value of the other variable. It is explained that the object of any algebraic method is to produce one equation with one variable. A second example is produced in which addition or subtraction does not help us directly. Equivalent equations are created by multiplying by appropriate constants. These equations can then be added (or subtracted, if that works) to get a single equation with one variable. A third complicated example is presented and solved. The original two equations involve fractional numerical coefficients which can be eliminated by multiplying each equation by a common denominator. The result is two equations which are solved as in example two. Cy and Lenny realize that an extremely large Cartesian plane is not the answer for solving a system of equations, and Lenny is quite upset that he must repaint the room. Objectives After viewing this program, and completing several suggested questions and exercises, students will be able to • realize that an algebraic method of solving two equations with two variables will produce the roots of a system of equations, no matter how complicated these roots may be; • realize that the algebraic method of elimination by addition or subtraction is directed toward producing one equation with one variable, which can be easily solved; • use this method of elimination to solve systems of simultaneous equations with two variables; • solve systems of linear equations which have fractional coefficients; solve word problems resulting in two equations with two variables. Before-Viewing Activities 2. Solve algebraically. 1. Solve the following systems, using the graphing approach. (a)x+2y= 7 3x-2y=-3 (b)2x+y=-2 3x+4y=2 (c) 5x - 3y = 15 -2x + 3y = -6 2. Estimate the roots of the following systems, using the graphing approach. (a)2x+3y=4 4x-y= 1 (b) x + y = -1 x-2y-4 After-Viewing Activities 1. Solve the following systems of equations using the algebraic method of elimination by addition or subtraction. (a) 3x + 2y = 13 x+2y=7 (b) 5x - 3y = 17 2x+3y=11 (c)5a+2b=8 4a - 3b = -12 (d) 2m + 7n = 2 5m+3n=-2 3. Write full solutions for the following word problems. A full solution includes opening statements to introduce the variables, the translation of the words into two equations with two variables, the solution of the system, and final statements which answer the problem. (a) The sum of two real numbers is 17 and their difference is 7. Find the numbers. (b) Three times one number, plus twice a second number is 58. Ten times the first number less three times the second number is -29. Find the numbers. (c) A bag contains a total of 20 dimes and quarters. The total value of the coins is $3.95. How many dimes are in the bag? (d) Tickets to a school play cost $5.00 for adults and $3.00 for students. A total of 249 tickets is sold for $961.00. How many adults and how many students purchased tickets? (e) Mary is five years older than her brother Bill. Eleven years ago, she was twice as old as her brother. How old are they now? PROGRAM 5 SUBSTITUTION AND COMPARISON Program Synopsis When Lenny wants to substitute Brand X checkered paint for the ceiling, Cy gets the idea that substitution might lead to another algebraic method for solving a linear system. After the method of elimination by addition or subtraction is reviewed, the substitution method is introduced. One of the equations is chosen from a system of two equations. This equation is solved for y in terms of x, and this expression for y is then substituted for y in the other equation. Once again, we obtain a single equation with one variable which is easy to solve. This value for x is then substituted to find the value for the other variable. This method is reviewed in a second example. It is shown that an equation can be solved for x in terms of y, and the expression for x substituted for x in the other equation. A fourth method, the comparison method, is presented. In this approach, both equations are solved for one of the variables in terms of the other variable. These two expressions, for example y, are then equated resulting in one equation with one variable. Finally, we look at a system of three equations with three variables. To solve this system, we are looking for values for x, y, and z which will satisfy all three equations at the same time. The elimination method is used to reduce the system to two equations with two variables; then to one equation with one variable. The solution to this equation is then back-substituted to find the value of a second variable, and then the value of the third variable. Cy and Lenny discuss the problem of graphing an equation with three variables. They suggest that a 3-dimensional plane would be handy (sort of like they had before they repainted the room, much to Lenny's chagrin). Objectives Before-Viewing Activities After viewing this, program and working on suggested problems, students will be able to 1. Solve the following systems using the elimination method. (a) 4x - 3y = 5 • . solve an equation with two variables for one of the variables in terms of the other; • solve a system of simultaneous equations with two variables using the substitution method; • solve a system of equations using the comparison method; • recognize when it may be most appropriate to choose a particular one of the three available algebraic methods; • use the elimination method to solve a system of three equations with three variables. 3x + 2y - 8 (b) 5x + 2y = 14 4x+7y=-5 2. Solve the following equations for y in terms of x. (a) 4x + y = 3 (b)3x-2y+6=0 (c)3(x-2)+5(2y+5)=9 3. Given y = 2x - 3, solve the following equations for x. (a) 3x + 4y = 5 (b) 2x - y = 9 2. Solve the following using the comparison method. (a) x+3y=-8 x-2y=12 (b) 3x + y =10 5x-y=-10 (c)-3x-5y+4=0 (c) 2x + 3y = 5 4. (a) The sum of two numbers is 188. Three times the smaller number is 24 less than the larger number. Find the numbers. (b) The sum of the digits of a 2-digit number is 10. If the number is doubled, the result is 28 more than the number with the digits reversed. Find the number. (c) A parking meter contains a total of $11.20 made up from a total of 154 nickels and dimes. How many dimes are there? 3x-2y=14 3. Solve the following using any algebraic method of your choice. In each question, is one method preferable over the others? Why? (a) 3x + y = 0 4x-7y=-25 (b) 3x + 2y = 35 5x-3y=-5 After-Viewing Activities 1. Solve the following systems using the substitution method. (a) 3x + y = 13 (c) x+5y=-5 x-3y=11 2x+5y=13 (b)x+4y=9 3x-2y=-1 4. Solve the following systems with three variables. (a) x+3y+2z=15 3x-3y-2z=-11 2x+y+z=8 (c) 3x + 5y = -9 4x-3y=-12 (b) 4a-b+2c=14 3a+2b+3c=9 5a+b-7c=27 5. (a) A person who won $6000.00 in a lottery invests part at 8.5% per year, and the remainder at 7.5% per year. How much is invested at each rate if the total interest earned in one year is $480.00? (b) How many kilograms of 40% salt solution and 30% salt solution should be mixed in order to obtain 200 kg of 37% salt solution? PROGRAM 6 THE THIRD DIMENSION Program Synopsis The program opens as one of the alien characters looks around using 3-dimensional glasses, and expresses how impressed he is. A system with three equations with three variables is solved by first selecting a pair of the equations and, using elimination by addition/subtraction, eliminating one of the variables. Another pair of equations, from the three, is then selected and the same variable is eliminated. We are left with two equations with two variables which can be easily solved. Once the values of these two variables are found, they can be substituted into one of the original equations to find the value of the third variable. The solution is then expressed as an ordered triple. The 3-dimensional coordinate system is introduced and the locations of two ordered triples are found. When these points are joined in 3-space, the result is a line. It is explained, with accompanying illustrations, that two lines in 3-space may intersect at one point, or they may be parallel with no point of intersection. A third situation is possible as well. In this case the two lines are not parallel, and do not intersect. They are called skew lines. The graph of a linear equation with three variables is discussed. Ordered triples are found and listed in a table of values. When these points are graphed, the result is a flat surface or plane. The intercepts of the plane are also discussed. If two planes are not parallel, the intersection is a line. When a third plane is introduced, it may tersect this line at one point. This is the solution of the three equations with three variables with which we started. It is also shown that three planes can be situated in other situations. One result gives a line as the solution, and another placement shows that there is no intersection and no solution to the linear system. The program ends as Cy and Lenny "morph" back into chairs as the class is set to resume. Objectives After viewing this program and completing several suggested exercises and questions, students will be able to • construct a 2-dimensional representation of a 3-dimensional space; • locate the positions of ordered triples in 3space; • recognize that a linear equation with three variables represents a plane in 3-space: • find the intercepts of a plane, and use these to draw a representation of the plane; • recognize that the solution of a linear system with three variables is the intersection of three planes, and that these three planes may intersect in a single point, a line, or not at all. Before-Viewing Activities 1. Find the x- and y-intercepts of the following lines. (a) 2x + 3y = 9 (b) 3x-5y+8=0 (c) 4x = 7 (d) 3y -12 = 0 2. Solve the following systems of three equations with three variables. (a) 2x + y + 2z =15 (b) Locate the points A(1, 2, 4), B(2, -3, 5), and C(0, -2,-3). 2. (a) Locate the points D(1, 2, 3) and E(-2, -1, 4). (b) Draw the line DE. (c) Draw a second line which is parallel to DE. (d) Draw a line which is not parallel to DE and which does not intersect DE. 3. Find the x-, y-, and z-intercepts of the following planes. 3x+y-3z=-9 (a) 2x+y+3z=6 4x+y+5z=32 (b) 3x-2y-2z=-6 (c) 4x+2y-z=4 (b) 2x + 3y + 4z = -20 3x - 2y + 2z = -1 4x + 5y + 5z = -26 3. Solve. 2x-y=6 4x-2y=8 Are these lines parallel or coincident? 4. Solve. -3x+6y-9=0 x-2y=-3 Are these lines parallel or coincident? After-Viewing Activities 1. (a) Construct a diagram of a 3-dimensional space (on your 2-dimensional page). 4. On separate graphs, sketch the planes defined in (2). 5. On a plain page, draw the following: (a) two planes which intersect in a line (b) two planes which are parallel (c) three planes which are parallel (d) three planes which intersect at a single point (e) three planes which intersect in a single line