MFM2P Foundations of Mathematics Unit 3 Lesson 11

Transcription

1 The Line Lesson

2 MFMP Foundations of Mathematics Unit Lesson Lesson Eleven Concepts Introduction to the line Using standard form of an equation Using y-intercept form of an equation x and y intercept Recognizing positive, negative, zero and undefined slopes Using the rise and the run of a given line to find its slope Using a pair of coordinates of a line to calculate slope Equation of a Line The equation of a line has basic forms. One is the y-intercept form and the other is called standard form. y-intercept form takes the form y mx + b Standard form looks like Ax +By + C 0 Y-intercept Form It is called y-intercept form because we can use the y-intercept in the equation to help us understand and graph the equation. y mx + b m value is the slope of the line. b s value is the spot on the y axis where the line intercepts the y axis. Example State the slope and y-intercept for the following equations. a. y x + 6 b. y x Solution a. y x + 6 rise slope m y-intercept b +6 run Copyright 00, Durham Continuing Education Page of 7

3 MFMP Foundations of Mathematics Unit Lesson b. y x rise slope m y-intercept b - run Example Graph the following equations using the slope and y-intercept. a. y x + b. y x Solution First plot the y-intercept. rise a. y x + m - run and b + Starting at the y-intercept go down and right then down again and right one again and so on Then draw a line through the points plotted to create your line. y x + Copyright 00, Durham Continuing Education Page of 7

4 MFMP Foundations of Mathematics Unit Lesson b. y x y x Support Questions. State the slope and y-intercept of the equations below, then properly graph and label the equations using the slope and y-intercept. a. y x + b. y x c. y x + 7 d. y x Slope Slope describes the steepness of a line or line segment; the ratio of the rise of a line or line segment to its run. Slope is either one of the following four types: a) positive b) negative c) no slope d) undefined Copyright 00, Durham Continuing Education Page of 7

5 MFMP Foundations of Mathematics Unit Lesson The formula for determining slope is: rise slope. run Slope between two points on a Cartesian Plane The slope of line segment can be determined if two ordered pairs of the line segment are known. The following formula is used: Example slope y x y x z a. Find the slope of the in line containing the coordinate pairs A(7,) and B(-9,7). b. Find the slope of the in line containing the coordinate pairs A(-,) and B(,6). Solutions ( x,y ) ( x,y ) a. (7, ) and (-9, 7) slope y x y x (7) () ( 9) (7) The s in the first ordered pair denotes the x and y values of that term and the s in the second ordered pair denotes the x and y values of that term. Substitute the values into the equation for slope. 6 Fraction needs to be simplified. Therefore the slope is. Copyright 00, Durham Continuing Education Page of 7

6 MFMP Foundations of Mathematics Unit Lesson b. (-, ) and (, 6) slope y x y (6) () x () ( ) Therefore the slope is 7. Support Questions. Find the slope of the in line containing the ordered pairs. a. A(,-), B(-,6) b. C(,6), D(7,) c. E(0,), F(,) d. G(7,), H(,-) e. I(,), J(,-) f. L(,7), M(,-) g. N(,-6),P(,-6). Find the missing value in the ordered pair. a. (,-), (x,6); slope - b. (,), (7,y); slope c. (x,), (,-); slope - d. (8,y), (-,-). Show by substitution which of the order pairs satisfies that equation. a. y x + (-,-), (,-), (-,7), (,-0) b. y x + (9,), (8,-), (,-7), (, -8) Finding the x intercept and the y intercepts The x intercept is where a line intersects or hits the x axis and the y intercept is where that same line intersects the y axis. Copyright 00, Durham Continuing Education Page 6 of 7

7 MFMP Foundations of Mathematics Unit Lesson Example Find the x and y intercepts for the equation x + y 0. Solution To find the x intercept: Set y 0 y (0) 0. Since this equals zero we can just ignore the y part of the equation. Therefore we ignore the y portion of the equation and solve for x. x 0 x + 0+ x x x This is the x intercept. Where the line x + y -0 hit the x-axis. To find the y intercept: Set x 0 Therefore we ignore the x portion of the equation and solve for y. y 0 y + 0+ y y y x (0) 0. Since this equals zero we can just ignore the x part of the equation. This is the y intercept. Where the line x + y -0 hit the y-axis. Support Questions. Find the x and y intercepts for each of the following equations then graph the line using the intercepts. a. x y b. x + 6y 0 c. x y Find the x and y intercepts for each of the following equations. a. y x + b. y x 7 Copyright 00, Durham Continuing Education Page 7 of 7

8 MFMP Foundations of Mathematics Unit Lesson Key Question #. State the slope and y-intercept of the equations below, then properly graph and label the equations using the slope and y-intercept. ( marks) a. y x + b. y x + 6 c. y x d. y x +. Find the slope of the line containing the ordered pairs. (7 marks) a. A(,-), B(-,7) b. C(,6), D(6,) c. E(,), F(0,) d. G(8,), H(,-) e. I(-,), J(,-) f. L(,7), M(,-) g. N(,-),P(,-6). Find the missing value in the ordered pair. (8 marks) a. (, 9), (x, ); slope b. (6, ), (, y); slope - c. (, y), (7, ); slope d. (x, -), (8, -6) -. Show by substitution which of the order pairs satisfies that equation. ( marks) a. y x + (-,-), (,-), (-,7), (,-0) b. y x 8 (9,), (8,-), (,-7), (, -8). Find the x and y intercepts for each of the following equations then graph the line using the intercepts. (9 marks) a. x + y 0 b. x y 0 0 c. x 6y Find the x and y intercepts for each of the following equations. ( marks) a. y x 9 b. y x + 7. Explain, with words and examples, when graphing equations of lines is easier using the y-intercept/slope method versus the x and y intercept method and when graphing equations of lines is easier using the x and y intercept method versus y- intercept/slope method. ( marks) Copyright 00, Durham Continuing Education Page 8 of 7

9 Parallel and Perpendicular Lines Lesson

10 MFMP Foundations of Mathematics Unit Lesson Lesson Twelve Concepts Converting equation of a line from y-intercept form to standard form Recognizing and graphing parallel slopes Recognizing and graphing perpendicular slopes Parallel Lines Lines that are parallel to each other have the same slope. Example Which of the following equations are parallel to each other? Solution y x y x + y x y x + y x and y x + are parallel to each other because they have the same slope of x. y x + y x Copyright 00, Durham Continuing Education Page 0 of 7

11 MFMP Foundations of Mathematics Unit Lesson Support Questions. Write an equation that is parallel to each equation below. a. y x + 7 b. y x + c. y x + 6 d. y x 7. Circle then graph the pair of equations that are parallel. y x + y x + y x y x 6 y x 7. Write each equation in y-intercept form then state which equations are parallel. a. x y b. 9 x 6y c. 6 x y + 0 d. x 8y 8 0 Perpendicular Lines Lines that are perpendicular to each other will have inverted slopes with opposite polarity. Example Which of the following equations are perpendicular to each other. y x + y x + y x + 6 y x Copyright 00, Durham Continuing Education Page of 7

12 MFMP Foundations of Mathematics Unit Lesson Support Questions. Write an equation that is perpendicular to each equation below. a. y x + 7 b. y x + c. y x + 6 d. y x 7. Circle then graph the pair of equations that are perpendicular. y x + y x + 7 y x y x 6 y x + 6. Write each equation in y-intercept form then state which equations are perpendicular. a. x 6y b. x + y c. 9 x 6y + 0 d. x + 6y 8 0 Copyright 00, Durham Continuing Education Page of 7

13 MFMP Foundations of Mathematics Unit Lesson Key Question #. Write an equation that is parallel to each equation below. ( marks) a. y x b. y x + c. y x d. y x. Circle then graph the pair of equations that are parallel. ( marks) y x y x + y x y x 6 y x +. Write each equation in y-intercept form then state which equations are parallel. ( marks) a. x + 0y b. 7 x y + 0 c. x + 6y 0 d. x 6y 8 0. Write an equation that is perpendicular to each equation below. ( marks) a. y x b. y x + c. y x d. y x. Circle then graph the pair of equations that are perpendicular. ( marks) y x y x + 7 y x y x 6 y x + 6. Write each equation in y-intercept form then state which equations are perpendicular. ( marks) a. 7 x y + 0 b. x + 0y c. x y + 0 d. x + y 0 7. It is possible to have two equations with different numerically different slopes yet are parallel. How is this possible? Explain and provide an example to illustrate your understanding. ( marks) 8. Does the y-intercept in an equation play any role in determining whether or not two lines are parallel or perpendicular? Explain and provide an example to illustrate your understanding. ( marks) Copyright 00, Durham Continuing Education Page of 7

14 Solving System of Equations: Graphically Lesson

15 MFMP Foundations of Mathematics Unit Lesson Lesson Thirteen Concepts Graphing linear equations Checking whether a coordinate pair satisfies and equation Finding the coordinates of the point of intersection of two equations Recognizing the meaning of the point of intersection of a linear system of equations Solving System of Equations by Graphing Example Find the point of intersection of these two equations. (Solve the system of equations graphically) y x + y x + 7 Solution The point of intersection is (-, ) which solves the system of equations. There is only one coordinate pair that satisfies both equations. The coordinates of the point of intersection are this coordinate pair. Check to see if the coordinate pair satisfies both equations. y x ( ) ( ) y x+ 7 ( ) ( ) Copyright 00, Durham Continuing Education Page of 7

16 MFMP Foundations of Mathematics Unit Lesson Support Questions. Match the coordinate pair with the equation it satisfies and prove by substitution. a. y x + 7 b. y x + c. y x + 6 d. y x 7 (, ), (-, ), (0, -), (-, -). Solve for the system of equations by graphing. (Find the point of intersection) y x + a. b. y x + y x + 6 y x + c. y x y x +. Two halls are being considered for a wedding reception. Hall one charges $0 per guest and hall two charges $ per guest and a fixed cost of $00. a. Write an equation for each hall. b. Graph both equations. c. Find the coordinates of the point of intersection. d. What do the number values in the coordinate pair represent? e. When is it best to rent hall one? f. When is it best to rent hall two?. To mow a lawn company A charges $ per hour and Company B charges a flat fee of $70 per season and $ per hour. a. Write an equation for each company. b. Graph both equations. c. Find the coordinates of the point of intersection. d. What do the number values in the coordinate pair represent? e. When is it best to use company A? f. When is it best to use company B? Copyright 00, Durham Continuing Education Page 6 of 7

17 MFMP Foundations of Mathematics Unit Lesson Key Question #. Match the coordinate pair with the equation it satisfies and prove by substitution. (8 marks) a. y x + b. y x + c. y x + 7 d. y x (, -), (, -), (, ), (-6, 0). Solve for the system of equations by graphing (Find the point of intersection). Prove by substitution that your answer is correct. (6 marks) y x + a. b. y x + y x y x + c. y x + y x. The graph below represents the cost to produce and sell sub sandwiches. Graph A represents the daily cost to produce subs and graph B represents the daily income from the sub sales. ( marks) a. Describe what the point of intersection represents. b. How many subs must be sold before there is a profit? Explain. Copyright 00, Durham Continuing Education Page 7 of 7

18 MFMP Foundations of Mathematics Unit Lesson Key Question # (continued). A company makes and sells video games. Each game sells for $8. The income, C dollars, from the sale of x games is given by C 8x. The cost to produce the games is given by C x (8 marks) a. Graph both equations. b. Find the coordinates of the point of intersection. c. What do the number values in the coordinate pair represent? d. How many games must be sold before there is a profit?. To prune small trees Company A charges $ per hour and Company B charges a flat fee of $80 per season and $ per hour. ( marks) a. Write an equation for each company. b. Graph both equations. c. Find the coordinates of the point of intersection. d. What do the number values in the coordinate pair represent? e. When is it best to use company A? f. When is it best to use company B? 6. Explain why solving a system of equations graphically can be difficult? Use y x and y -x + as your example. ( marks) Copyright 00, Durham Continuing Education Page 8 of 7

19 Solving System of Equations Using Substitution Lesson

20 MFMP Foundations of Mathematics Unit Lesson Lesson Fourteen Concepts Isolating a variable Checking whether a coordinate pair satisfies and equation Finding the coordinates of the point of intersection of two equations using algebra Recognizing the meaning of the point of intersection of a linear system of equations Solving System of Equations with Substitution Example Solve the system of equations using substitution. y x + y x + 7 Solution Since y y and the y in the first given linear equation is equal to x + then we replace the y in the second equation with x +. y x + y x + 7 -x + x +7 Algebra is now used to solve for x. x+ x+ x+ x+ 7 x x x 6 x x Remember when using algebra keep the equation balanced at all times. Take the value of x and substitute it into either equation and solve for y. y y y x+ 7 ( ) + 7 Copyright 00, Durham Continuing Education Page 0 of 7

21 MFMP Foundations of Mathematics Unit Lesson So the point of intersection (solution) is (-, ). Check to see if the coordinate pair satisfies both equations. y x ( ) ( ) y x+ 7 ( ) ( ) Example Solve the system of equations using substitution. x y x + y 8 Solution Need to isolate x or y in either equation and then use substitution. x+ y 8 x x+ y x 8 y x 8 Since y yand the y in the first given equation is equal to x 8, then we replace the second equation with x 8. x y ( ) x x y Remember to use the x + x distributive law ( ) and ( 8) + 8 Algebra is now used to solve for x. x+ x+ 8 x + 8 x x 0 x 0 x Copyright 00, Durham Continuing Education Page of 7

22 MFMP Foundations of Mathematics Unit Lesson Take the value of x and substitute it into either equation and solve for y. x+ y 8 ( ) + y 8 + y y 8+ y So the point of intersection (solution) is (-, -). Check to see if the coordinate pair satisfies both equations. x + y 8 ( ) + ( ) x y ( ) ( ) 6+ Support Questions. Isolate y in each linear equation. a. x - y b. 6x +y c. 7x + y. Isolate x in each linear equation. a. x - y b. x +0y c. x + y 7. Solve for the system of equations by using substitution. Check your solution using substitution. y x a. b. y x + 8 y x y x c. y x 0 y x + 8 x y + d. e. x y + 8 x y x y + f. x y x + y 8 g. x y x + y 8 Copyright 00, Durham Continuing Education Page of 7

23 MFMP Foundations of Mathematics Unit Lesson. Two halls are being considered for a wedding reception. Hall one charges $0 per guest and hall two charges $0 per guest and a fixed cost of $00. a. Write an equation for each hall. b. Solve the system of equations using substitution. c. What do the number values in the coordinate pair represent? d. When is it best to rent hall one? e. When is it best to rent hall two? Key Question #. Isolate y in each linear equation. (6 marks) a. x +y b. x +y 8 c. 7x y. Isolate x in each linear equation. (6 marks) a. x +y b. x +8y 8 c. x 6y. Solve for the system of equations by using substitution. Check your solution using substitution. ( marks) y x + a. b. y x 6 y x + y x c. y x + y x + x y d. e. x y x y + x y f. x y 9 x + y 8 g. x y x + y 7. A company makes and sells video games. Each game sells for $8. The income, C dollars, from the sale of x games is given by C 8x. The cost to produce the games is given by C x Solve the system of equations using substitution. ( marks). To prune small trees Company A charges $6 per hour and Company B charges a flat fee of $ per season and $ per hour. a. Write an equation for each company. b. Solve the system of equations using substitution. c. What do the number values in the coordinate pair represent? d. When is it best to use company A? e. When is it best to use company B? Copyright 00, Durham Continuing Education Page of 7

24 Solving System of Equations Using Elimination Lesson

25 MFMP Foundations of Mathematics Unit Lesson Lesson Fifteen Concepts Solving system of linear equations using elimination Checking whether a coordinate pair satisfies and equation Finding the coordinates of the point of intersection of two equations using algebra Recognizing the meaning of the point of intersection of a linear system of equations Solving System of Equations with Elimination Example Solve the system of equations using elimination. y x + y x + 7 Solution What is important for elimination is to line up the x s with the x s ; the y s with the y s and the constants with the constants. What is needed next is to have the same coefficient (ignore the sign for now) for either the x s or the y s. y x + y x + 7 The y s in this example both have a coefficient of so the y s will be eliminated. If the coefficients of the variable being eliminated have the same sign then subtract each term. Copyright 00, Durham Continuing Education Page of 7

26 MFMP Foundations of Mathematics Unit Lesson If the coefficients of the variable being eliminated have the opposite sign then add each term. Algebra is now used to solve for x. 0 x 6 0+ x x+ x 6 x 6 x 6 x Take the value of x and substitute it into either equation and solve for y. y y y x+ 7 ( ) + 7 y x + 7 So the point of intersection (solution) is (-, ). Check to see if the coordinate pair satisfies both equations. y x ( ) ( ) y x+ 7 ( ) ( ) Example Solve the system of equations using elimination. x y 0 x + y 6 Copyright 00, Durham Continuing Education Page 6 of 7

27 MFMP Foundations of Mathematics Unit Lesson Solution With this question it does not matter whether the y s or the x are eliminated since there are no matching coefficients. Method. Eliminating the y s. Since the positive value of the coefficients of the y s are and then the lowest common multiple (LCM) of and is needed. LCM of and is 0. So in the first equation each term must be multiplied by so that the becomes a 0 and in the second equation each term must be multiplied by so the becomes a 0. (x y -0) becomes 0x 0y -0 and (x +y -6) becomes 0x +0y -90 so the two linear equations are now 0x 0y -0 0x + 0y -80 The signs of the 0 s are opposite so we add each of the terms to eliminate the y s. Solve for y: x+ y 6 ( ) + y y 6+ 8 y 8 y 8 y Copyright 00, Durham Continuing Education Page 7 of 7

28 MFMP Foundations of Mathematics Unit Lesson So the point of intersection (solution) is (-, -). Check to see if the coordinate pair satisfies both equations. x + y -6 x y -0 (-) + (-) -6 (-) (-) Method. Eliminating the x s. Since the positive value of the coefficients of the x s are and then the lowest common multiple (LCM) of and is needed. LCM of and is 60. So in the first equation each term must be multiplied by so that the becomes a 60 and in the second equation each term must be multiplied by so the becomes a 60. (x y -0) becomes 60x 0y -0 and (x +y -6) becomes 60x +0y -0 so the two linear equations are now 60x 0y -0 60x + 0y -0 The signs of the 60 s are the same so we subtract each of the terms to eliminate the x s. Copyright 00, Durham Continuing Education Page 8 of 7

29 MFMP Foundations of Mathematics Unit Lesson Solve for x: x+ y 6 ( ) x + 6 x 8 6 x x 8 x 8 x So the point of intersection (solution) is (-, -). Check to see if the coordinate pair satisfies both equations. x + y -6 x y -0 (-) + (-) -6 (-) (-) Support Questions. What is the LCM of each of the following pairs of numbers? a., b., 7 c., 8 d., 6 e.,. Multiply each of the following equations by. a. x - y b. x +0y c. x + y 7. For each of the system of linear equations given below state whether you would add or subtract each term to eliminate the x s. y x a. b. y x + 8 y x y x c. y x 0 y x + 8 x y + d. e. x y + 8 x y x y + 6x y f. 6x + y 8 g. x y x + y 8 Copyright 00, Durham Continuing Education Page 9 of 7

30 MFMP Foundations of Mathematics Unit Lesson Support Questions (continued). Solve for the system of equations by using elimination. Check your solution using substitution. y x + a. b. y x 6 y x + y x c. y x + y x + x y d. e. x y x y x + y 7. At a sale, each DVD movie is one price and each VHS movie is another price. Three DVDs and two VHS s cost $67. One DVD and three VHS s cost $8. Let d dollars represent the price of one DVD and let v dollars represent the price of one VHS. a. Write the equations for each of the two purchase combination. a. Solve the system of equations using elimination. b. What does one DVD cost? c. What does one VHS cost? Key Question #. What is the LCM of each of the following pairs of numbers? ( marks) a., b., 7 c., 8 d., 6 e.,. Multiply each of the following equations by. (6 marks) a. x - y b. x +0y c. x + y 7. For each of the system of linear equations given below state whether you would add or subtract each term to eliminate the x s. (6 marks) x + y 7 a. b. x y x y 0 x + y 8 c. x + y x + y 8 x + y 8 d. e. x + y 6x y 6x + y 8 f. x + y x + y Copyright 00, Durham Continuing Education Page 0 of 7

31 MFMP Foundations of Mathematics Unit Lesson Key Question # (continued). Solve for the system of equations by using elimination. Check your solution using substitution. ( marks) x + y 7 a. b. x y x y 0 x + y 8 c. x + y 7x + y 8 x + y 8 d. e. x + y 6x y x + y 8 f. x + y x + y. Three soccer balls and one volleyball cost $. Two soccer balls and three volleyballs cost $0. Let s dollars represent the price of one soccer ball and let v dollars represent the price of one volleyball. ( marks) a. Write the equations for each of the two combinations. b. Solve the system of equations using elimination. c. What does one volleyball cost? d. What does one soccer ball cost? 6. Describe one advantage and one disadvantage of each of the three methods of solving systems of linear equations. (graphing, substitution and elimination) (6 marks) Copyright 00, Durham Continuing Education Page of 7

32 MFMP Foundations of Mathematics Support Question Answers Answers to Support Questions Lesson Eleven. a. m -; b b. m ; b - c. m ; b 7 d. m ; b -. a. ( ) (6) (6) () m b. m () ( ) 6 6 () (7) c. () () m d. (0) () () ( ) m (7) () 7 6 () ( ) 8 e. m undefined f. () () 0 (7) ( ) 9 m () () ( 6) ( 6) 0 g. m 0 () (). a. b. ( ) (6) 8 m () (x) x ()( 8) ( x) 8 + x x x x x c. d. () ( ) m (x) () x ()() (x ) x + x + 0 x 0 x x y + 0 y y y () (y) y m () (7) 6 ()( y) ( 6) y y y y y (y) ( ) y + m (8) ( ) 0 ()(y + ) 0() y + 0 Copyright 00, Durham Continuing Education Page of 7

33 MFMP Foundations of Mathematics Support Question Answers. a. (, ); () b. (8, ); (8) + 6 +, 8 ; ( ) ( ) 8 8. a. y int y y y 8 y 8 y (0, ) x int x x x 8 x 8 x (, 0) b. y int 6y 0 6y y 6y 6 6 y x int x 0 x + 0+ x x x (0, ) (, 0) Copyright 00, Durham Continuing Education Page of 7

34 MFMP Foundations of Mathematics Support Question Answers c. y int y + 0 y + 0 y y y x int x + 0 x + 0 x 6. a. b. y int y + y x int 0 x + 0 x + x (0, ) (, 0) y y int (0, ) x int 0 x ( x) 7 x x ( ) (, 0) Lesson Twelve. a. y x ±any number b. y x ±any number c. y x ±any number d. y x ±any number. y x + and y x Copyright 00, Durham Continuing Education Page of 7

35 MFMP Foundations of Mathematics Support Question Answers. a. b. x y x x y + 7 x y + 7 x y x 7 y x 7 y x 7 7 y x+ 9x 6y x 9x 6y x 6y + 6 9x 6y x 6 6y 9x 6 6y 9x y x+ c. d. 6x y + 0 6x 6x y + 0 6x y + 6x y + 6x y 6x y 6x y x+ x 8y 8 0 x x 8y 8 0 x 8y 8 x 8y 8+ 8 x+ 8 8y x+ 8 8y x y x Therefore equations b and c are parallel to one another and so are equations a. and d.. a. y -x ±any number b. y x ±any number c. y x ±any number d. y x ±any number. y x + and y x + Copyright 00, Durham Continuing Education Page of 7

36 MFMP Foundations of Mathematics Support Question Answers 6. a. b x 6y x x 6y x 6y + 7 x 6y x 7 6y x 7 6y x y x+ 6 x+ y x x+ y x y + 6 x y x 6 y x 6 c. d. 9x 6y + 0 9x 9x 6y + 0 9x 6y + 9x 6y + 9x 6y 9x 6y 9x 6 6 y x+ x+ 6y 8 0 x x+ 6y 8 0 x 6y 8 x 6y 8+ 8 x+ 8 6y x+ 8 6y x y x+ Therefore equations a. and b. are perpendicular to one another and so are equations c. and d. Lesson Thirteen. a. b. c. d. (, ) y x (, ) y x + () + + (0, ) (, ) y x 7 y x + 6 (0) 7 ( ) Copyright 00, Durham Continuing Education Page 6 of 7

37 MFMP Foundations of Mathematics Support Question Answers. a. b. c.. a. y 0x ; y x + 00 b. and c. d. The same cost for which ever hall chosen. e. Hall one when more than 67 guests. f. Hall two when less than 67 guests. Copyright 00, Durham Continuing Education Page 7 of 7

38 MFMP Foundations of Mathematics Support Question Answers. a. y x ; y x + 70 b. and c. d. The same cost for which ever company chosen. e. Company A when more than hours. f. Company B when less than hours. Lesson Fourteen. a. b. c. x y x x y x y x 6x+ y 6x 6x+ y 6x y 6x 7x+ y 7x + 7x+ y + 7x y x y 6x y + 7x y + x y x. a. b. c. x y x y + y + y x+ 0y x + 0y 0y 0y x 0y x+ y 7 x + y y 7 y x 7 y x + y x 0y x 7 y x y x 7+ y. a. y x y x+ 8 x x+ 8 x+ x x+ x+ 8 x 8 x + 8+ x x x y x y () y 6 y point of intersection (, ) Copyright 00, Durham Continuing Education Page 8 of 7

39 MFMP Foundations of Mathematics Support Question Answers check y x () 6 y x + 8 () b. y x y x x x x x x x x x + + x 8 x 8 x y y y x point of intersection (, ) check y x () y x c. y x 0 y x+ 8 x 0 x+ 8 x+ x 0 x+ x+ 8 6x 0 8 6x x 8 6x x y x 0 y () 0 y 9 0 y point of intersection (, ) Copyright 00, Durham Continuing Education Page 9 of 7

40 MFMP Foundations of Mathematics Support Question Answers check y x 0 () y x+ 8 () d. check x y + x y + 8 y + y + 8 y y + y y + 8 y + 8 y + 8 y y y x y + x + x 0 point of intersection (0, ) x y x y ( ) Copyright 00, Durham Continuing Education Page 0 of 7

41 MFMP Foundations of Mathematics Support Question Answers e. x y x y + y y + y + y y + y + 7y + 7y + 6 7y 6 7y y check x y x ( 8) x x 0 point of intersection (0, 8) 0 y 0 ( 8) x y + 0 ( 8) f. x y y x y x+ x+ y 8 y x 8 y x+ y x 8 x+ x 8 x+ x+ x+ x 8 x + 8 x + 8 x 0 x 0 x x+ y 8 ( ) + y 8 + y y 8+ y point of intersection (, ) Copyright 00, Durham Continuing Education Page of 7

42 MFMP Foundations of Mathematics Support Question Answers check x + ( ) 8 ( ) + ( ) x y ( ) ( ) 6+ g. x y y x+ y x x+ y 8 y x x+ (x ) 8 x+ 8x 8 8 9x 8 8 9x x 6 9x 6 9x x x y () y 8 y 8 8 y 8 y 6 y 6 y 6 point of intersection (,6) check x y () (6) 8 6 x + y 8 + (6) Copyright 00, Durham Continuing Education Page of 7

43 MFMP Foundations of Mathematics Support Question Answers. a. C 0x C 0x +00 b. 0x 0x+ 00 0x 0x 0x 0x+ 00 0x 00 0x x 0 C 0(0) C 00 (0, 00) c. With 0 guests both halls charge the same amount of $00. d. Hall one with less than 0 guests. e. Hall two with more than 0 guests Lesson Fifteen. a. b. 8 c. 0 d. e. (x y ). a. b. x y (x + 0y ) x + 0y c. ( x + y 7) x + 9y. a. add b. sub c. add d. sub e. sub f. sub g. add Copyright 00, Durham Continuing Education Page of 7

44 MFMP Foundations of Mathematics Support Question Answers. a. - y x+ y x 6 0 x x x 9 x x y x+ ( ) y + y 6+ y check y x+ ( ) + 6+ y x 6 ( ) 6 6 b. ( ) solution, y x+ - y x 0 x + 0+ x x+ x+ x x x check y x y x+ ( ) + + y y y x ( ) ( ) solution, Copyright 00, Durham Continuing Education Page of 7

45 MFMP Foundations of Mathematics Support Question Answers c. - y x+ y x x x y x+ y 0+ y check y x+ 0+ y x+ ( ) ( ) solution 0, d. - x y x y 0 y y 0 y check x y ( ) 0 0 ( ) x 0 x 0 x ( ) solution,0 x y ( ) 0 0 Copyright 00, Durham Continuing Education Page of 7

46 MFMP Foundations of Mathematics Support Question Answers e. x y x+ y 7 Multiply x y by and it becomes 6x y 8 Multiply x+ y 7 by and it becomes 6x 9y - 6x y 8 6x 9y y check y y x y () x x x + + x 6 x 6 x x y () () 6 x+ y 7 () + () ( ) solution,0 Copyright 00, Durham Continuing Education Page 6 of 7

47 MFMP Foundations of Mathematics Support Question Answers. a. d +v 67 and d +v 8 b. d + v 67 d + v 8 Multiply nd equation by - d + v 67 d + 9v 7v 77 7v v d + v 8 ( ) d + 8 d + 8 d + 8 d c. one DVD costs $ d. one VHS costs $ ( ) solution, Copyright 00, Durham Continuing Education Page 7 of 7