Student Outomes Students know that two equations in the form of ax + y = and a x + y = graph as the same line when a = = and at least one of a or is nonzero. a Students know that the graph of a linear equation ax + y =, where a,, and are onstants and at least one of a or is nonzero, is the line defined y the equation ax + y =. Lesson Notes Following the Exploratory Challenge is a Disussion that presents a theorem aout the defining equation of a line (page 367) and then a proof of the theorem. The proof of the theorem is optional. The Disussion an end with the theorem and in plae of the proof, students an omplete Exerises 4 8 eginning on page 369. Whether you hoose to disuss the proof or have students omplete Exerises 4 8, it is important that students understand that two equations that are written differently an e the same and their graph will e the same line. This reasoning eomes important when you onsider systems of linear equations. In order to make sense of infinitely many solutions to a system of linear equations, students must know that equations that might appear to e different an have the same graph and represent the same line. Further, students should e ale to reognize when two equations define the same line without having to graph eah equation, whih is the goal of this lesson. Students need graph paper to omplete the Exploratory Challenge. Classwork Exploratory Challenge/Exerises 1 3 (0 minutes) Students need graph paper to omplete the exerises in the Exploratory Challenge. Students omplete Exerises 1 3 in pairs or small groups. Exploratory Challenge/Exerises 1 3 1. Sketh the graph of the equation 9x + 3y = 18 using interepts. Then answer parts (a) (f) that follow. 9(0) + 3y = 18 3y = 18 y = 6 The y-interept is (0, 6). 9x + 3(0) = 18 9x = 18 x = The x-interept is (, 0). Date: 11/19/14 364
a. Sketh the graph of the equation y = 3x + 6 on the same oordinate plane.. What do you notie aout the graphs of 9x + 3y = 18 and y = 3x + 6? Why do you think this is so? The graphs of the equations produe the same line. Both equations go through the same two points, so they are the same line.. Rewrite y = 3x + 6 in standard form. y = 3x + 6 3x + y = 6 d. Identify the onstants a,, and of the equation in standard form from part (). a = 3, = 1, and = 6 e. Identify the onstants of the equation 9x + 3y = 18. Note them as,, and. = 9, = 3, and = 18 f. What do you notie aout a a,, and? a = 9 = 3, 3 = 3 = 3, and 1 = 18 6 = 3 Eah fration is equal to the numer 3.. Sketh the graph of the equation y = 1 x + 3 using the y-interept and the slope. Then answer parts (a) (f) that follow. a. Sketh the graph of the equation 4x 8y = 4 using interepts on the same oordinate plane. 4(0) 8y = 4 8y = 4 y = 3 The y-interept is (0, 3). 4x 8(0) = 4 4x = 4 x = 6 The x-interept is ( 6, 0).. What do you notie aout the graphs of y = 1 x + 3 and 4x 8y = 4? Why do you think this is so? The graphs of the equations produe the same line. Both equations go through the same two points, so they are the same line. Date: 11/19/14 365
. Rewrite y = 1 x + 3 in standard form. y = 1 x + 3 (y = 1 x + 3) y = x + 6 x + y = 6 1( x + y = 6) x y = 6 d. Identify the onstants a,, and of the equation in standard form from part (). a = 1, =, and = 6 e. Identify the onstants of the equation 4x 8y = 4. Note them as,, and. = 4, = 8, and = 4 f. What do you notie aout a a,, and? a = 4 = 4, 1 = 8 = 4, and = 4 6 = 4 Eah fration is equal to the numer 4. 3. The graphs of the equations y = x 4 and 6x 9y = 36 are the same line. 3 a. Rewrite y = x 4 in standard form. 3 y = x 4 3 (y = x 4) 3 3 3y = x 1 x + 3y = 1 1( x + 3y = 1) x 3y = 1. Identify the onstants a,, and of the equation in standard form from part (a). a =, = 3, and = 1. Identify the onstants of the equation 6x 9y = 36. Note them as,, and. = 6, = 9, and = 36 d. What do you notie aout a a,, and? a = 6 = 3, = 9 3 = 3, and = 36 1 = 3 Eah fration is equal to the numer 3. Date: 11/19/14 366
e. You should have notied that eah fration was equal to the same onstant. Multiply that onstant y the standard form of the equation from part (a). What do you notie? x 3y = 1 3(x 3y = 1) 6x 9y = 36 After multiplying the equation from part (a) y 3, I notied that it is the exat same equation that was given. Disussion (15 minutes) Following the statement of the theorem is an optional proof of the theorem. Below the proof are Exerises 4-8 (eginning on page 369) that an e ompleted instead of the proof. What did you notie aout the equations you graphed in eah of Exerises 1 3? In eah ase, the graphs of the equations are the same line. What you oserved in Exerises 1 3 an e summarized in the following theorem: MP.8 THEOREM: Suppose a,,,,, and are onstants, where at least one of a or is nonzero, and one of a or is nonzero. (1) If there is a nonzero numer s so that = sa, = s, and = s, then the graphs of the equations ax + y = and x + y = are the same line. () If the graphs of the equations ax + y = and x + y = are the same line, then there exists a nonzero numer s so that = sa, = s, and = s. The optional part of the disussion egins here. We want to show that (1) is true. We need to show that the graphs of the equations ax + y = and x + y = are the same. What information are we given in (1) that will e useful in showing that the two equations are the same? We are given that = sa, = s, and = s. We an use sustitution in the equation x + y = sine we know what,, and are equal to. Then y sustitution, we have By the distriutive property Divide oth sides of the equation y s x + y = sax + sy = s. s(ax + y) = s. ax + y =. Whih means the graph of x + y = is equal to the graph of +y =. That is, they represent the same line. Therefore, we have proved (1). Date: 11/19/14 367
To prove (), we will assume that a, 0, that is, we are not dealing with horizontal or vertial lines. Proving () will require us to rewrite the given equations ax + y = and x + y = in slope-interept form. Rewrite the equations. ax + y = y = ax + y = a x + y = a x + x + y = y = x + a y = x + y = a x + We will refer to y = a x + a as (A) and y = x + as (B). What are the slopes of (A) and (B)? The slope of (A) is a a, and the slope of (B) is. Sine we know that the two lines are the same, we know the slopes must e the same. a = a When we multiply oth sides of the equation y 1, we have a = a. By the multipliation property of equality, we an rewrite a = a as a a =. Notie that this proportion is equivalent to the original form. What are the y-interepts of y = a x + a (A) and y = x + (B)? The y-interept of (A) is, and the y-interept of (B) is. Beause we know that the two lines are the same, the y-interepts will e the same. We an rewrite the proportion. = = Saffolding: Students may need to see the intermediate steps in the rewriting of a = a a as =. a Date: 11/19/14 368
Using the transitive property and the fat that a = a and a = =. Therefore, all three ratios are equal to the same numer. Let this numer e s. a = s, = s, = s =, we an make the following statement: We an rewrite this to state that = sa, = s, and = s. Therefore, () is proved. When two equations are the same, i.e., their graphs are the same line, we say that any one of those equations is the defining equation of the line. Exerises 4 8 (15 minutes) Students omplete Exerises 4 8 independently or in pairs. Consider having students share the equations they write for eah exerise while others in the lass verify whih equations have the same line as their graphs. Exerises 4 8 4. Write three equations whose graphs are the same line as the equation 3x + y = 7. Answers will vary. Verify that students have multiplied a,, and y the same onstant when they write the new equation. 5. Write three equations whose graphs are the same line as the equation x 9y = 3 4. Answers will vary. Verify that students have multiplied a,, and y the same onstant when they write the new equation. 6. Write three equations whose graphs are the same line as the equation 9x + 5y = 4. Answers will vary. Verify that students have multiplied a,, and y the same onstant when they write the new equation. 7. Write at least two equations in the form ax + y = whose graphs are the line shown elow. Answers will vary. Verify that students have the equation x + 4y = 3 in some form. Date: 11/19/14 369
8. Write at least two equations in the form ax + y = whose graphs are the line shown elow. Answers will vary. Verify that students have the equation 4x + 3y = in some form. Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: We know that when the graphs of two equations are the same line it is eause they are the same equation in different forms. We know that even if the equations with the same line as their graph look different (i.e., different onstants or different forms) that any one of those equations an e referred to as the defining equation of the line. Lesson Summary Two equations define the same line if the graphs of those two equations are the same given line. Two equations that define the same line are the same equation, just in different forms. The equations may look different (different onstants, different oeffiients, or different forms). When two equations are written in standard form, ax + y = and x + y =, they define the same line when a a = = is true. Exit Tiket (5 minutes) Date: 11/19/14 370
Name Date Exit Tiket 1. Do the graphs of the equations 16x + 1y = 33 and 4x + 3y = 8 graph as the same line? Why or why not?. Given the equation 3x y = 11, write another equation that will have the same graph. Explain why. Date: 11/19/14 371
Exit Tiket Sample Solutions 1. Do the graphs of the equations 16x + 1y = 33 and 4x + 3y = 8 graph as the same line? Why or why not? No, in the first equation a = 16, = 1, and = 33, and in the seond equation a = 4, = 3, and = 8. Then, a = 4 16 = 1 4, = 3 1 = 1, ut 4 = 8 33 = 8 33. Sine eah fration does not equal the same numer, then they do not have the same graph.. Given the equation 3x y = 11, write another equation that will have the same graph. Explain why. Answers will vary. Verify that the student has written an equation that defines the same line y showing that the frations a a = = = s, where s is some onstant. Prolem Set Sample Solutions Students pratie identifying pairs of equations as the defining equation of a line or two distint lines. 1. Do the equations x + y = and 3x + 3y = 6 define the same line? Explain. Yes, these equations define the same line. When you ompare the onstants from eah equation you get a = 3 = 3, 1 = 3 = 3, and 1 = 6 = 3. When I multiply the first equation y 3, I get the seond equation. Therefore, these equations define the same line. (x + y = )3 3x + 3y = 6. Do the equations y = 5 x + and 10x + 8y = 16 define the same line? Explain. 4 Yes, these equations define the same line. When you rewrite the first equation in standard form you get y = 5 x + 4 (y = 5 x + ) 4 4 4y = 5x + 8 5x + 4y = 8. When you ompare the onstants from eah equation you get a = 10 5 =, = 8 =, and 4 = 16 8 =. When I multiply the first equation y, I get the seond equation. Therefore, these equations define the same line. (5x + 4y = 8) 10x + 8y = 16 Date: 11/19/14 37
3. Write an equation that would define the same line as 7x y = 5. Answers will vary. Verify that the student has written an equation that defines the same line y showing that the frations a a = = = s, where s is some onstant. 4. Challenge: Show that if the two lines given y ax + y = and x + y = are the same when = 0 (vertial lines), then there exists a non-zero numer s, so that = sa, = s, and = s. When = 0, then the equations are ax = and x =. We an rewrite the equations as x = and x = a a. Beause the equations graph as the same line, then we know that and we an rewrite those frations as a = a =. These frations are equal to the same numer. Let that numer e s. Then a = sa and = s. a = s and = s. Therefore, 5. Challenge: Show that if the two lines given y ax + y = and x + y = are the same when a = 0 (horizontal lines), then there exists a non-zero numer s, so that = sa, = s, and = s. When a = 0, then the equations are y = and y =. We an rewrite the equations as y = and y =. Beause the equations graph as the same line, then we know that their slopes are the same. We an rewrite the proportion. = = These frations are equal to the same numer. Let that numer e s. Then and = s. = s and = s. Therefore, = s Date: 11/19/14 373