What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases Lesson Objective:

Transcription

1 What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases Lesson Objective: Length of Activity: Students will continue work with solving systems of equations using the equal values method when equations are not in y-form and learn to identify systems that represent the same line or parallel lines (that is, systems that have no solution). One day (approximately 45 minutes) Core Problems: 1 through 3 Materials: Lesson Overview: Suggested Lesson Activity: Closure: (7-10 minutes) None This lesson concludes the introductory work with systems of linear equations. It considers the case where one of the equations is not in y = form, which serves as a foreshadowing of the substitution method. Students will also examine the special cases of linear systems where the equations represent the same (concurrent) line and when the system has no solution (parallel lines). The lesson opens with a problem that uses all of the topics in this section of the text. Students analyze a situation, write equations to represent the relationships, then use the equal values method to solve the system of equations. The values in the problem have been chosen so that students must also deal with fractions in the equations. Problem 2 asks students to consider what to do to solve a system when one of the problems is not in y = form. The focus here is on the meaning of equality. The first part of the problem asks students to devise a way to use the equal values method to solve the system. One way to do this is to solve for y in the second equation. The second part of the problem focuses on equality and asks students to see that they can use this idea to solve the equation without having to have both equations in y = form. Problem 3 introduces students to linear systems that represent parallel and concurrent lines. The questions have students examine the algebraic structure of the equations as well as graph them to see the geometry of their graphs. The remaining problems offer additional practice with solving systems. You may want to spend some time focusing on ways to identify the special cases without actually solving. The problems in the lesson involve the basics of solving linear systems while incorporating work with special cases and fraction coefficients.

2 What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases You have been introduced to systems of linear equations that are used to represent various situations. You used the equal values method to solve systems algebraically. Today you will determine the relationship within a system of equations by examining them carefully. 1. Sara has agreed to help with her younger sister s science fair experiment. Her sister planted string beans in two pots. She is using a different fertilizer in each pot to see which one grows the tallest plant. At present, plant A is 4 tall and grows 2 3 per day. Plant B is 9 tall and grows 1 per day. If the plants 2 continue growing at these rates, in how many days will the two plants be the same height? Which plant will be tallest in six weeks? [ y = 2 3 x + 4, y = 1 2 x + 9, x = 30 days. Plant A. ] 2. Jesus applied for a job. The application process required that he take a test of his math skills. One problem on the test was a system of equations with one of the equations not in y = mx + b form. The two equations are shown below. y = 2 5 x! 5 3x + 2y = 7 Work with your team to find a way to solve the equations using the equal values method. [ Change second equation to y = 7 2! 3 2 x, x = and y =! ]

3 3. Using the equal values method can lead to messy fractions. Sometimes that cannot be avoided. But some systems of equations can be solved by simply examining them. This approach is called solving by inspection. Consider the two cases below. Case I: 3x + 2y = 2 Case II: 2x! 5y = 3 3x + 2y = 8 4x! 10y = 6 a. Compare the left sides of the two equations in Case I. How are they related? [ equal ] b. Use the equal values method for solving a system of equations, write a relationship for the two right sides of the equations in Case I, and explain your result. [ 2 = 8, which is never true, so there is no solution. ] c. Graph the two equations in Case I to confirm your result for part (b) and to see how the graphs of the two equations are related. [ See graph at right; lines are parallel. ] d. Recall that a coefficient is a number multiplied by a variable and a constant is a number alone. Compare the coefficients of x, the coefficients of y, and the two constants in the equations in Case II. How is each pair of integers related? [ Each corresponding value in the second equation is twice that of the first (or, the values in the first are half that of the second). ] e. Half of your team should multiply the coefficients and constant in the first equation in Case II by 2 and then solve the system using the equal values method. The other half of your team should divide all three values in the second equation in Case II by 2 and then solve using the equal values method. Compare the results from each method. What does your result mean? [ The result is that the two equations are multiples of each other (either 2 or 1 2 ), so they are the same. They both represent the same line. ] f. Graph the two equations in Case II to confirm your result in part (e). [ See graph at right; same line. ]

4 ETHODS AND MEANINGS MATH NOTES Solutions to a System of Equations A solution to a system of equation gives a value of the each variable that makes both equations true. For example, when 4 is substituted for x and 5 is substituted for y in both equations at right, both equations are true. So x = 5 and y = 5 or 4, 5 ( ) is a solution to this system of equations. When the two equations are graphed ( 4, 5) is the point of intersection. Some systems of equations have no solutions or infinite solutions. Consider the examples at right. Notice that the equal values method would yield 3 = 4 which is never true. When the lines a graphed they are parallel and so the system has no solution. However, in the third set of equations, the second equation is just the first equation multiplied by two. Therefore, the two lines are really the same line and have infinite solutions. System with one solution intersecting lines x! y =!1 2x! y = 3 System with no solution parallel lines: x + y = 3 x + y = 4 System with infinite solutions coinciding lines: x + y = 3 2x + 2y = 6 5. The graph at right contains the lines for y = x + 2 and y = 2x! 1. a. Using the graph, what is the solution to this system? [ (3, 5) ] b. Solve the system algebraically to confirm your answer to part (a). y = x + 2 y = 2x! 1

5 6. Change each equation below into y = mx + b form. [ a: y = 4x! 3, b: y = x + 3, c: y =! 3 2 x + 6, d: y =! 2 3 x + 2 ] a. y! 4x =!3 b. 3y! 3x = 9 c. 3x + 2y = 12 d. 2(x! 3) + 3y = 0 7. Mailboxes Plus sends packages overnight for $5 plus $0.25 per ounce. United Packages charges $2 plus $0.35 per ounce. Mr. Molinari noticed that his package would cost the same to mail using either service. How much does his package weigh? [ 30 ounces ] 8. Graph each equation below on the same set of axes and label the point of intersection with its coordinates. [ ( 2, 1) ] y = 2x + 3 y = x GETTING IN SHAPE Frank weighs 160 pounds and is on a diet to gain two pounds a week so that he can make the football team. John weighs 208 pounds and is on a diet to lose three pounds a week so that he can be on the wrestling team in a lower weight class. a. If Frank and John can meet these goals with their diets, when will they weigh the same, and how much will they weigh at that time? [ In 9.6 weeks, they will both weigh pounds. ] b. Clearly explain your method.

6 10. Aimee thinks the solution to the system below is ( 4, 6). Eric thinks the solution is (8, 2). [ They are both correct. The lines coincide. ] 2x! 3y = 10 6y = 4x! 20 a. Is Aimee correct? b. Is Eric correct? c. What do the answers to (a) and (b) tell you about the lines in the problem? 11. Consider these two equations: y = 3x! 2 y = 4 + 3x a. Graph both equations on the same set of axes. b. Solve this system using the Equal Values Method. [ no solution ] c. Explain how the answer to part (b) agrees with the graph you made in part (a). [ There is no solution to the system of equations because the two lines do not intersect. ] 12. Find the solution for each system of equations below, if a solution exists. If there is not a single solution, explain why not. Be sure to check your solution, if possible. [ a: no solution, b: infinite solutions because the lines coincide ] a. x + 4y = 2 x + 4y = 10 b. 2x + 4y =!10 x + 2y =!5

7 What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases You have been introduced to systems of linear equations that are used to represent various situations. You used the equal values method to solve systems algebraically. Today you will determine the relationship within a system of equations by examining them carefully. 1. Sara has agreed to help with her younger sister s science fair experiment. Her sister planted string beans in two pots. She is using a different fertilizer in each pot to see which one grows the tallest plant. At present, plant A is 4 tall and grows 2 3 per day. Plant B is 9 tall and grows 1 per day. If the plants 2 continue growing at these rates, in how many days will the two plants be the same height? Which plant will be tallest in six weeks? 2. Jesus applied for a job. The application process required that he take a test of his math skills. One problem on the test was a system of equations with one of the equations not in y = mx + b form. The two equations are shown below. y = 2 5 x! 5 3x + 2y = 7 Work with your team to find a way to solve the equations using the equal values method. 3. Using the equal values method can lead to messy fractions. Sometimes that cannot be avoided. But some systems of equations can be solved by simply examining them. This approach is called solving by inspection. Consider the two cases below. Case I: 3x + 2y = 2 Case II: 2x! 5y = 3 3x + 2y = 8 4x! 10y = 6 a. Compare the left sides of the two equations in Case I. How are they related?

8 b. Use the equal values method for solving a system of equations, write a relationship for the two right sides of the equations in Case I, and explain your result. c. Graph the two equations in Case I to confirm your result for part (b) and to see how the graphs of the two equations are related. d. Recall that a coefficient is a number multiplied by a variable and a constant is a number alone. Compare the coefficients of x, the coefficients of y, and the two constants in the equations in Case II. How is each pair of integers related? e. Half of your team should multiply the coefficients and constant in the first equation in Case II by 2 and then solve the system using the equal values method. The other half of your team should divide all three values in the second equation in Case II by 2 and then solve using the equal values method. Compare the results from each method. What does your result mean? f. Graph the two equations in Case II to confirm your result in part (e).

9 ETHODS AND MEANINGS MATH NOTES Solutions to a System of Equations A solution to a system of equation gives a value of the each variable that makes both equations true. For example, when 4 is substituted for x and 5 is substituted for y in both equations at right, both equations are true. So x = 5 and y = 5 or 4, 5 ( ) is a solution to this system of equations. When the two equations are graphed ( 4, 5) is the point of intersection. Some systems of equations have no solutions or infinite solutions. Consider the examples at right. Notice that the equal values method would yield 3 = 4 which is never true. When the lines a graphed they are parallel and so the system has no solution. However, in the third set of equations, the second equation is just the first equation multiplied by two. Therefore, the two lines are really the same line and have infinite solutions. System with one solution intersecting lines x! y =!1 2x! y = 3 System with no solution parallel lines: x + y = 3 x + y = 4 System with infinite solutions coinciding lines: x + y = 3 2x + 2y = 6 5. The graph at right contains the lines for y = x + 2 and y = 2x! 1. a. Using the graph, what is the solution to this system? b. Solve the system algebraically to confirm your answer to part (a). y = x + 2 y = 2x! 1

10 6. Change each equation below into y = mx + b form. a. y! 4x =!3 b. 3y! 3x = 9 c. 3x + 2y = 12 d. 2(x! 3) + 3y = 0 7. Mailboxes Plus sends packages overnight for $5 plus $0.25 per ounce. United Packages charges $2 plus $0.35 per ounce. Mr. Molinari noticed that his package would cost the same to mail using either service. How much does his package weigh? 8. Graph each equation below on the same set of axes and label the point of intersection with its coordinates. y = 2x + 3 y = x GETTING IN SHAPE Frank weighs 160 pounds and is on a diet to gain two pounds a week so that he can make the football team. John weighs 208 pounds and is on a diet to lose three pounds a week so that he can be on the wrestling team in a lower weight class. a. If Frank and John can meet these goals with their diets, when will they weigh the same, and how much will they weigh at that time? b. Clearly explain your method.

11 10. Aimee thinks the solution to the system below is ( 4, 6). Eric thinks the solution is (8, 2). 2x! 3y = 10 6y = 4x! 20 a. Is Aimee correct? b. Is Eric correct? c. What do the answers to (a) and (b) tell you about the lines in the problem? 11. Consider these two equations: y = 3x! 2 y = 4 + 3x a. Graph both equations on the same set of axes. b. Solve this system using the Equal Values Method. c. Explain how the answer to part (b) agrees with the graph you made in part (a). 12. Find the solution for each system of equations below, if a solution exists. If there is not a single solution, explain why not. Be sure to check your solution, if possible. a. x + 4y = 2 x + 4y = 10 b. 2x + 4y =!10 x + 2y =!5