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TRS099 Diagnostic Unit4 Notes - Graphing.pdf

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Department
Business
Course
BUS 100
Professor
Marla Spergel
Semester
Fall

Description
Math Skills for Business- Full Chapters 86 U4 Full Chapter 11- Graphing in the Coordinate Plane 11.1 Introduction to Graphing A graph may be regarded as a pictorial representation of data. A graph may be drawn manually on paper, or by using software on a graphing calculator or computer. Graphs, especially of functions are mostly drawn on a rectangular co-ordinate plane. A rectangular co-ordinate plane is formed by the intersection of two number lines. One, on the East-West directions is called the horizontal axis, and the other on the North-South directions is called the vertical axis. The horizontal and vertical axis, make up the co-ordinate axis, and divide the co-ordinate plane into four regions called quadrants (fig. 1). fig. 1 vertical axis Second Quadrant First Quadrant QQQuadrant 0 horizontal axis Third Quadrant Fourth Quadrant QQQuadrant Why a Co-ordinate Plane? Because a point on a co-ordinate plane is also identified as the ordered pair of numbers (horizontal co-ordinate, vertical co-ordinate), and an ordered pair of numbers interpreted as (horizontal co-ordinate, vertical co-ordinate) is identified as a point on a co-ordinate plane. The horizontal co-ordinate is the number on the horizontal axis, and the vertical co-ordinate is the number on the vertical axis associated with the point on the co-ordinate plane. Math Skills for Business- Full Chapters 87 In (fig. 2) the points A, B, C, D are also indentified by the attached pair of numbers. The ordered pairs (2, 3), (-5, 4), (-3.5, -6), (3,-2.5) are the points attached. fig. 2 y 5*B(0,5) 4 *(-5,4) 3 *(2,3) 2 *A(3,2) 1 -6A(--50) -4 -3 -2 -1 0 1 2 3 4 5 x o -1 r -2 i z *(3,-2.5) o -3 n t -4 a l -5 *C(4,-5) *(-3.5,-6) -6 NOTE: In fig.2, the letter x is representing the horizontal axis and in such a case the horizontal axis becomes the x-axis, and similarly the vertical axis is the y-axis. It should be noted that the axis can be assigned any names. Graph of Functions: What is a function? For our purpose, it is a set of instructions and procedures which takes a number called the INPUT NUMBER to produce another number called the Math Skills for Business- Full Chapters 88 OUTPUT NUMBER. The set of instructions and procedures are given either by words (e.g. XX), or by algebraic equations (e.g. YY). Therefore the function is also given by the set of ordered pair of numbers (INPUT NUMBER, Corresponding OUTPUT NUMBER) of the function for all possible input numbers. Function is: INPUT NUMBER Instructions OUTPUT NUMBER and Procedure machine And also the set of ordered pair of numbers (INPUT NUMBER, OUTPUT NUMBER) for each input number and its corresponding output number. The ordered pair of numbers (INPUT NUMBER, OUTPUT NUMBER) is also a point on a co-ordinate plane whose horizontal axis is the INPUT NUMBERS and the vertical axis is the OUTPUT NUMBERS. For such a coordinate plane the horizontal axis is called the INPUT-axis, and the vertical axis the OUTPUT-axis. The graph of a function is always on such a coordinate plane, and it is all the points with coordinates (INPUT NUMBER, corresponding OUTPUT NUMBER). How is the graph of a function constructed? 1. Choose a letter (e.g. t) or word (e.g. distance) to represent the INPUT numbers, and a different letter (e.g. q) or (e.g. amount) to represent the OUTPUT numbers if they are not given. From the examples, the horizontal axis is then the t-axis and the vertical axis the q-axis. 2. Make a table (of values) with the headings INPUT (eg t), OUTPUT (eg q), and (INPUT, OUTPUT) {eg (t, q)}. From the possible INPUT numbers choose about 11 different numbers if the function if not linear and 4 numbers if it is a linear function. For each INPUT number chosen find the corresponding OUTPUT number and the ordered pair of numbers (INPUT, OUTPUT). 3. With the INPUT and OUTPUT numbers from (2) as a guide choose a scale for the INPUT (horizontal) and OUTPUT (vertical) axis and draw both axis. Math Skills for Business- Full Chapters 89 4. Plot the points for the ordered pair of numbers found in (2). If all the points are on a straight line join them by a straight edge (e.g. ruler) and extend in both directions. Otherwise join the points by a curve in order of magnitude of the INPUT numbers, and extend in both directions. 5. Give a title which identifies the function or the application to your graph. All the functions in the following examples are linear and so the graphs are straight lines. By Euclid we need only two points, but we add an extra or two as a check. Example 1 Dora works a night shift in a warehouse and earns $10/hour plus $2.00 premium bonus per hour. Draw a graph for Dora’s earnings given the amount of time worked. Use your graph to estimate how much she will earn for working 8 hours. Solution The INPUT is the amount of time worked and OUTPUT is amount earned, since earning is for work done. Amount of time worked in hours is represented by t and the amount earned (wages) in dollars by w. The horizontal axis is therefore the t-axis, and the vertical axis the w-axis. The graph may be titled “The Wages of Dora”. From the question Dora’s hourly earnings = $10 + $2 = $12. On the assumption that Dora’s hourly earnings is unchanged, with hours worked we make the calculations: t in h w in $ (t,w) 1 12 (1,2) 2 24 (2,24) 4 48 (4,48) 5 60 (5,60) Math Skills for Business- Full Chapters 90 From the two arrows on the graph Dora earns $96 in 8 hours. Example 2: A computer technician charges $30 for diagnosing computer problems and $20/hour for fixing the problems. How much will she charge for work done on a computer which took her 2 hours after diagnosis to fix? (Find solution by graph). Solution: The INPUT number is the amount of time in hours, t, after diagnosis it takes technician to fix problem. The OUTPUT number is the amount in dollars, a, the technician charges for diagnosis and fixing the problem. The horizontal axis is therefore the t-axis, and the vertical axis the a-axis. The graph may be titled “Fee for Diagnosis and Repair”. On the assumption that the diagnosis fee and the hourly rate is unchanged by the amount of time it takes to diagnose and/or fix, we make the following calculations: t in h a in $ (t, a) 0 30 + 20(0) = 30 (0,30) 1 30 + 20(1) = 50 (1,50) 2 30 + 20(2) = 70 (2,70) 3 30 + 20(3) = 90 (3,90) Math Skills foFees for Diagnosis and Repair91 From the two arrows on the graph, she will charge $80 for Diagnosis and 2 hours repair time. Example 3 The cost in dollars ‘C’ of preparing meals for ‘m’ number of people is given by the equation: C = 8n + 25 a) Draw a graph of the equation. b) From the graph, what is the cost of preparing meals for 9 people? c) From the graph, how many people can be fed by $121? Solution: ‘C’ is expressed in terms of ‘m’, and by convention the m-values are the INPUT and the C-values are the OUTPUT numbers. Therefore the horizontal axis is the m-axis, and the vertical axis is the C-axis. ‘m’, is the number of people, so a value of m cannot be negative. Within this constraint choose any four numbers for m, to make the Table of Values. The title of the graph is the equation: C = 8m + 25. Math Skills for Business- Full Chapters 92 a) n C = 8n + 25 (n, C) C = 8n + 25 0 8(0) + 25 = 25 (0,25) D 1 8(1) + 25 = 33 (8,33) 3 8(3) + 25 = 49 (3,49) 5 8(5) + 25 = 65 (5,65) B b) From the arrows B on the graph it will cost about $96 to feed 9 people. c) From the arrows D on the graph, $121 can feed about 12 people. Comment: From the equation if n = 9, then C = 8(9) +25 = 97. This $97 is the exact cost of meals for 9 people. The $96 from the graph is an approximate solution. In many instances graphs give only approximate solutions. Example 4: The demand per week of children toys is 30,000 when the price is $20 and 20,000 when the price is $25 each. When the price increased to $30 demand fell to 10,000. This is summarized in the table below: Unit Price in $: p 20 25 30 Quantities Demanded per week: q 30,000 20,000 10,000 a) Graph the relationship between unit price and the quantities demanded per week. b) Use the graph to find out the demanded quantities when the price fell to $10.00 c) How much will be the gross income of the toy producer when 30,000 were demanded at $20.00 a toy? Solution The p-values are the INPUT numbers, and the q-values are the OUTPUT numbers, therefore the horizontal axis is the p-axis and the vertical axis is the q-axis. It is Math Skills for Business- Full Chapters 93 assumed that the graph of demand per week (q) against unit price is a straight line. From the table in the question the table of values of the graph is made: a) Unit price and Demand of Toys p q (p, q) 20 30 000 (20, 30 000) 25 20 000 (25, 20 000) 30 10 000 (30, 10 000) b) From the arrows on the graph, at $10 per unit price the demand is 50, 000 toys. c) Gross Income in $ = 20 (30,000) = 600,000. Gross Income is $600,000. Comments on: Example 1: (i) The points (3, 36) and (7, 84) are on the graph. (Please check) The difference of the vertical coordinates of these two points = 84 – 36 = 48. The difference of the corresponding horizontal coordinates = 7 – 3 = 4. For any other two points on the graph, similar calculation will give the same result 12. (Pick any two points on the graph, to check the claim.) (ii) The mathematical relationship between the vertical coordinates w, and the horizontal coordinate t of points on the graph is given by the equation: w = 12t. Example 2: (i) The points (1.5, 60) and (4, 110) are on the graph. (Please check) Math Skills for Business- Full Chapter94 For any other two points on the graph, similar calculation will give the same result 20. (Pick any two points on the graph, to check the claim.) (ii) The mathematical relationship between the vertical coordinates, a, and the horizontal coordinates t of points on the graph is given by the equation: a = 20t + 30. Example 3: The graph is for the equation: C = 8n + 25. Pick any two points on the graph and check that: Example 4: (i) The points (13, 4400) and (23, 24000) are on the graph. (Please check) For any other two points on the graph, similar calculation will give the same result -2000. (Pick any two points on the graph, to check the claim.) (ii) The mathematical relationship between the vertical coordinates q, and the horizontal coordinate p is given by the equation: q = -2000p + 70 000. 11.2 Linear Relations A linear relation between an OUTPUT variable (e.g. y) and an INPUT variable (e.g. x) is geometrically a straight line graph, and mathematically an equation usually of form y = mx + c. So examples 1, 2, 3, 4 are linear relations, and y, x, m, and c are: Examples y x m c Equation 1 w t 12 0 w = 12t. 2 a t 20 30 a = 20t + 30 3 C n 8 25 C = 8n + 25 (given) 4 q p -2000 70 000 q = -2000p + 70 000 Math Skills for Business- Full Chapters 95 The ‘m’ is called the slope or the gradient of the line graph (in analogy with an incline). It is also referred to as the rate of change of the OUTPUT value to a unit change of the INPUT value (in analogy to applications as in the examples). (0, b) is called the y-intercept, and is the point of intersection of the line graph and the vertical axis. These two values the slope m and the point (0, b) describe both the graph and its equation. They can be used to draw the graph and/or write the equation of the graph in the slope intercept form; y = mx + b . Interpreting Linear Graphs: If the graph trends upwards from left to right, it indicates that as x increases y increases, and as x decreases y decreases and the Slope is positive. (eg 1, 2, 3). A classic example is Price and Demand ‘curve’. If the graph trends downwards from left to right, it indicates that as x increases y decreases, and as x decreases y increases and the Slope is positive. eg 4). A classic example is Price and Supply ‘curve’. If the graph is parallel to the x-axis (horizontal line), then an increase or decrease in x causes no change in y that is y has a fixed value, and the Slope is zero. If the graph is parallel to the y-axis (vertical line), then x has a fixed value that causes y to take all values, and the Slope is undefined. The Slope, m, given the line: This is also the first step in finding the equation of a line or line through given points. Math Skills for Business- Full Chapters 96 Example 5 The table below shows the hourly earnings (in dollars) for daycare workers in a wood manufacturing company in Ontario from 2001 to 2005. Calendar year x 2001 2002 2004 2005 Hourly rate y in $ 14.20 14.50 15.10 15.40 Find the rate of change in the hourly rate per year of the workers from 2001 to 2005. Solution: Plot the points. If they lie on the same straight line then the rate of change for any two periods of time is the same, and is equal to the slope of the line. (2001, 14.20) and (2002, 14.50) are on the line. Therefore the rate of change in hourly earnings between 2001 and 2005 is $0.30 / year. Example 6 Find the slope of the line containing (-1, 0) and (3, 4). Solution It is given that P = (-1, 0) and P = (3, 4) are on the line. Slope m = = = = = 1 Slope of the line, m = 1. Math Skills for Business- Full Chapters 97 Example 7 Show that if a graph is a horizontal line, then its slope is 0. Solution Horizontal line is parallel to the x- axis. Any point on the line is of form (h, k), h is any number, k is a fixed number and (0, k) is the y- intercept. Example 8 Show that if a graph is a vertical line, then its slope is undefined. Solution Vertical line is parallel to the y-axis. A point on it is of form (k, v), k fixed, and v is any number. Math Skills for Business- Full Chapters 98 11.3 Graph of the linear equation: ax + by + c = 0 In general, a linear equation of x and y is of form: Ax + By + C = 0. We can also solve this to give: y = mx + b −A −C −C m = B , c = B . m is the slope, (0,b) the y-interceAt, 0) the x − intercept of the graph of the equation Ax + By + C = 0. The equation of form y = mx + c is called the slope-intercept form (for obvious reasons). Take Note: From the equation the slope, the y-intercept, and the x-intercept of the graph can be determined before it is drawn. From this information it can be deduced whether the graph is horizontal (if A = 0), vertical (if B = 0), trends upwards (if m is positive) or downwards (if m is negative) from left to right. The information can also be used to draw the graph of the equation. Solutions of equation and points on its graph 1. A solution (h, k) of an equation is the coordinates of a point on its graph. 2. The coordinates of any point on a graph, is a solution of its equation. 11.4 Drawing Graph of Linear Equation The graph of a linear equation is a straight line. By Euclid a straight line is defined by any two points on it because there is only one straight line that passes through two points. Combined with the first statement above, any two solutions of the linear equation can be used to draw its graph. That is, plot the two points and draw a straight line extended at both ends to pass through the two points. Methods of Solution of Linear Equation: i. Guess and verify. For most equations it is inefficient and impossible for some. ii. Substitute any number for x in the equation and solve to find corresponding y. iii. Substitute any number for y in the equation and solve to find corresponding x. Math Skills for Business- Full Chapters 99 y-intercept: Is the point of intersection of the graph and the y-axis. A point on the y-axis is of form (0, k). So to find the y-intercept, 0 is substituted for x in method (ii). x-intercept: Is the point of intersection of the graph and the x-axis. A point on the x- axis is of form (h, 0). So to find this point, 0 is substituted for y in method (iii). It is often most efficient to combine methods (ii) and (iii) and/or the intercepts. Example 9: (Find and use intercept to draw graph). a) Find the x- and y-intercepts of the line 4x -2y = 8 b) Graph the equation. Solution a) The x-intercept is a point of form (h, 0) on the graph of the equation. So x = h, y = 0 is a solution of the equation. Substitute in equation and solve for h. Therefore the x-intercept is the point (2, 0). The y-intercept is a point of form (0, k) on the graph of the equation. So x = 0, y = k is a solution of the equation. Substitute in equation and solve for k. Therefore the y-intercept is the point (0, -4). b) Plot the two points (2, 0) and (0, -4) and draw a straight line through them. Math Skills for Business- Full Chapters 100 Comment: In practice to find the y- intercept, simply substitute x=0 in the given linear equation and solve for y. Similarly to find the x-intercept, simply substitute y=0 in the given linear equation and solve for x. In applications it is sometimes of interest to know the output when virtually no input (input = 0) has been made. For example the price of an item when supply is almost zero, and this is deduced from the y-intercept. Equally of interest is the level of input that produces no output (output = 0). For example the level of supply at which the price of an item collapses (price = 0), and this is deduced from the x-intercept. Example 10: (Find the slope and y-intercept from equation) Write the equation 2x – 3y = 6 in the standard form y = m x + b. i) Identify the slope and the y-intercept. ii) Use (i) to plot graph of equation. Solution ii) The point (0, -2) is on the graph, plot it. From the value of the slope, 2x – 3y = 6 a second point on the graph is: (0 + 3, -2 + 2) = (3, 0), plot it. Draw a line to pass through the two points. This line is the graph of the equation Math Skills for Business- Full Chapters 101 Example 11: (Graph by Slope & y-intercept, or two solutions method) Graph the equation 3x + y = 2. Solution Method 1: Put equation in the form y = m x + b, to find the slope and y-intercept. Plot the two points (0, 2) and (1, -1). 3x + y = 2 Draw a straight line to pass through the two points. This straight line is the graph. Plot the two points (0, 2) and (2, -4). Draw a straight line to pass through the two points. This straight line is the graph. Math Skills for Business- Full Chapters 102 3x + y = 2 11.5 Given a Straight line Graph find the Equation Given the slope and any point on the straight line graph, a second point can be found. A straight line graph is defined by any two points (Euclid), and so also by a slope and a point. Except a straight line graph is given on a coordinate axis, the straight line would therefore be defined by two points, or a slope and a point. Methods: 1. The slope of a straight line is the same for any two points on the line. 2. y = mx + b (m is slope, (0, b) is y-intercept) is the standard form of the equation of a straight line. The given two points, or slope and point are used to find m and b. Example 12 Find the equation of the line which contains the point (0, 5) and has a slope of 2. Solution: Method 1: Equation of line slope m, containing the point (h, k) is: (i) y – k = m(x – h) . From the question: m = 2, (h, k) = (0, 5). So substitute m = 2, h = 0, k = 5 in (i) Math Skills for Business- Full Chapters 103 Method 2: Standard equation is: (i) y = mx + b, m is the slope, (0, b) the y-intercept. From the question: m = 2, (0, b) = (0, 5). So substitute m = 2, b = 5 in (i). y = 2x + 5 is the equation of the line. Example 13 Write the equation of the line containing the points (-4, -5) and (-2, 4). Solution: Given: Two points (-3, -1) and (2, 4) on a line. Find: The equation of the line. From the two points the slope m of the line can be found. The question now becomes: find the equation of a line of slope m, and containing (-4, -5) or (-2, 4). Method 1: Equation of line slope m, containing the point (h, k) is: (i) y – k = m(x – h) . 4 − 5 9 From the two points (-4, -5) and (-2, 4) on the line the slop− m − ⇒ m = , and 2− 4 2 9 9 using (h, k) = (-4, -5). Substituting in (i) ⇒ y - 5 (x − 4)⇒ y + 5 = (x + 4)⇒ 2 2 9 9 9 9 y + 5 = 2 x +2 × 4 ⇒ y + 5 = 2x +18 (subtract 5 from both sides)⇒ y =2 x +13 This is the equation of the line in standard form. To write equation in general form: multiply both sides by 2 ⇒ 2y = 9x + 26 ⇒ 2y − 9x − 26 = 0 Method 2: Standard equation is: (i) y = mx + b, m is the slope, (0, b) the y-intercept. 4 − 5 9 From the two points (-4, -5) and (-2, 4) on the line, the sl−2 − 4=⇒ m = 2 then 9 substitute this value for m in (i) ⇒(ii)2 x + b. The value of b is such that for any point (h, k) 9 9 on the line k = h + b. Using (h, k) = (-2, 4)⇒ 4 = (−2) + b ⇒ 4 = − 9 + b 2 2 9 (add 9 to both sides)⇒13= b. Substitute b = 13 in (ii) ⇒ Equation of line is 2 x + 13. Math Skills for Business- Full Chapters 104 Method 3: Standard equation is: (i) y = mx + b, m is the slope, (0, b) the y-intercept. For any point (h, k) on the line; (ii) k = mh + b. The points (-4, -5) and (-2, 4) are on the line. For the point (-4, -5), substitute h = -4, and k = -5 in (ii) ⇒(iii) − 5 = − 4m + b For the point (-2, 4), substitute h = -2, and k = 4 in ⇒(iv) 4 = − 2m + b Solve equations (iii) and (iv) simultaneously to obtain m, and b. Substitute the values obtained in (i) for the equation of the line. Example 14 The table below shows the length of time in days Maya worked installing solar water heaters in houses and the number of houses with solar water heaters in Brampton. Time in days # of houses with swh i) Write an equation to represent the 2 8 relation between the number of houses 4 12 in Brampton with solar water heaters 6 16 and the length of time in days Maya 8 20 worked. 10 24 ii) How many houses in Brampton had solar water heaters before Maya’s work? Solution Use values in the table to plot a graph with ‘time in days’ represented by ‘t’ as input and ‘# of houses with swh’ represented by ‘s’ as output. The resulting graph on the right is a straight line. Therefore the relation between s and t is linear. It is the equation of the line. (Use any two points e.g. (2, 8) and (6, 16) and apply any of the methods in the previous examples to show that the equation of the line is: s = 2t + 4.) Math Skills for Business- Full Chapters 105 i) s = 2t + 4 is the relation between the number s of houses in Brampton with solar water heaters and the length of time in days t Maya worked installing solar water heaters. ii) From the equation: for t = 0, s = 4. So there were 4 houses in Brampton with solar water heaters before Maya started installing them. (Or from the graph the s-intercept is (0, 4) which leads to the same answer. The slope, intercepts, and equation of a line are very important concepts with wide application in business. So we advise the student to ensure that she/he has understood and mastered these concepts. 11.6 Review, Exercises and Assignments In question 1- 20, fill in the blanks with the appropriate words or answer true or false. 1. To calculate the rate of change in respect of y to x you need a change in ____ and a change in a ____ 2. The x-axis is the ____.That means that it does not depend on anything. 3. The y-axis is the ____. It depends on the independent variable. 4. In terms of change over time, the y-value is the ____ axis and the time is the ____ axis. Answer true or false for #5-#9: 5. The rate of change of the input value to the output value is the same as the slope. 6. The slope of a straight line graph is constant. 7. Calculation of the slope requires only subtraction and then division. 8. For any straight line the rate of change is constant. 9. Non-straight line graphs do not have constant slopes. 10. What two pieces of information are needed to write an equation of a straight line? ________ and ________. 11. A line that slants downward from left to right has a ____ slope. Math Skills for Business- Full Chapters 106 12. A vertical line has ________________ slope. 13. The formula y = mx + b is called ____________________________. 14. The y-intercept is ____. 15 The x-intercept is ____. 16. Generally, the dependable variable is usually called ____, and the dependent variable is also called ____. 17. A line that slants upward from left to right has a ____ slope. 18. With a slope and a co-ordinate point we can write an equation of a straight line. True or false? 19. A horizontal line has a slope of ____. 20. The slope may also be defined as the rise divided by the run. True or false? 21. The line segment AB has a slope of . If the co-ordinate of point A is (2, 5), the co-ordinates of point B could be which of the following: a) (6, 8) b) (5, 9) c) (-2, 2) d. (6, 2) 22. The graph of the equation 2x + 6y = 4 passes through point (k, 2). What is the value of k? 23. Point (-3, h) lies on the line whose equation is x -2y = -2. What is the value of h? 24. Which statement best describes the graph of x =4? a) It passes through the point (0, 4) b) It has a slope of 4. c) It is parallel to the y-axis. d) It is parallel to the x-axis. 25. What is the y-intercept of the graph of the line whose equation is y = - x +4? 26. If point (-1, 0) is on the line whose equation is y = 2x + b, what is the value of b? Math Skills for Business- Full Chapters 107 27. If the value of the input variable y increases as the value of the independent variable x increases, the graph of this relationship could be a) Horizontal line b) Vertical line c) One with a negative slope d)One with a positive slope. 28. The line 3x -2y = 12. Which of the following is correct? a). A slope of and a y-intercept of -6. b) A slope of - and a y-intercept of 6. 3) A slope of 3 and a y-intercept of -2 d) A slope of -3 and a y-intercept of -6 29. Write an equation of the line that has a slope of 3 and a y-intercept of -2 30. What is the x-intercept of the line 2x -3y =12? 31. The data in the table shows the cost (in $) of renting a bicycle in a resort by time (time rented in hours), including deposit. Hours (time rented) Cost ($) 2 15 5 30 8 45 a) What would be the equation of a line that fits the data? b) What is the amount of the deposit? Math Skills for Business- Full Chapters 108 32. The equation of line A is 5x + 6y = 3, and the equation of line B is 5x -6y =3. Which of the statement below about the two lines is true? a) Lines A and B have the same y-intercept. b) Lines A and B have the same slope. c) Lines A and B have the same x-intercept. 33. Write the equation of each line containing, a. A (3,2) and A (3, -4) b. B ( 2, 1) and B (-2, -3) c. C (-3, 5) and C (-2, 3) d. D (-1, -2) and D (3, -2) 34. Write the equation of each line that has the specified slope and contains the given points. a) Point (2,-3), m= 3 b) Point (0, -3), m = -1 c) Point (3, 5), m= - d) Point (5, 1), m = - 35. Graph the lines that passes through point a. (-2, -3) and has a slope b. (2,0) and has a slope -1 36. Calculate the slope and y-intercept of each of the equations below. a. 2x -3y = 6 b) x – 3y = 3 c) 4x -5y = 5 d) 4x + y =2 37.Find the slope of each line a. Contains points (-2, 3) and (4, 2) b. Contains points (2, 5) and (-2, 5) Math Skills for Business- Full Chapters 109 38. Using the equation y = - x + 2, find y when x =3 39. Graph 2x + 3y = 6 by using the x- and y- intercepts. 40. Intra Manufacturing Inc makes wood-burning heaters for rural and remote population. The production cost and quantities produced are linearly related. It costs $1500 to make 20 heaters and $2100 to make 30 heaters. a) If C is the cost of making x heaters, write an equation for this relations. b) Use the equation to calculate the cost of making 35 wood-burning heaters. 41.In 2000, the total gross sales of Ridgeview Electronic were $350000. The sales were $400000 in 2001 and $450,000 in 2002. Let s represent the total sales in x years. a. List the three co-ordinate points. b. Plot a graph relating s and x. 42.The Nanest Company produces solar panels. The analysis of the relations between the production cost ($c) of the company and its production quantities( Q), is described using the following linear equation: C= 1500 + 300Q a) Identify both the slope of the equation and its c-intercept. b) In a particular week, the production quantities were 500 solar panels, what was the production cost? c) If production cost was $45000, how many quantities were produced? d) What is the unit cost of producing 200 quantities? 43.For each of the following linear equations, list three solutions as ordered pairs in the form (x, y). Math Skills for Business- Full Chapters 110 a) y = x + 2 b) y = 2x c ) y = 3x – 1 44. a) Find four ordered pairs (a, b) for the formula, b = a + 3, for a =0, 1, 2 and 5. b) Draw the graph using the five ordered pairs in (a). c) Draw a line through the points. 45. The cost (in dollars) of producing a college newspaper is given by using the Formula, C = 2n + 400, where n is the number of copies printed and C the total cost. a. What is the slope of the formula? b. Draw a graph for the formula. (Hint: use n= 100, 200, 300, and 500) c. Use the graph to estimate the cost of producing thousand copies. 46. a) Draw the graph of a straight line passing through the points (0, 0) and (3, 6). b) Find the slope of the line. c) Write an equation for the line. 47. A certain car is expected to depreciate in value according to the equation, y = − 2000x + 40000, where y is the value of the car (in $) and x the age of the car (in years) a) Find the slope of the line and interpret its meaning. b) Find the y-intercept and explain what it means. 48. Write the equation of the line that: a) passes through the point (3, 4) and has slope -3 b) passes through the point (-1, -4) and has slope of . 49. What is the slope of the graph of the linear equation 5y – 10x – 15= 0? Math Skills for Business- Full Chapter111 50. The table below shows the enrolment of a daycare centre from 2002 through 2006. Year(x) Enrolment (y) 2002 14 2003 20 2004 22 2005 28 2006 37 a) Draw a graph for the above relations b) Between 2002 and 2006, enrolment increased by what percent? c) Which year had the lowest increase in enrolment? d) Which year had the highest increase in enrolment? 51. A straight line was created using the following data. x 0 1 2 3 y -6 -3 0 3 What are the x-and y-intercept for this line? 52. A and b are linearly related. The values of a and its corresponding values are given in the table below: a 0 1 2 b 2 4 6 Write a formula for this relation. Math Skills for Business- Full Chapters 112 53. Given the equation y = x – 2. a) What is the slope of this equation? b) Does the graph of this equation rise or fall from left to right? c) What is the y-intercept? d) What is the x-intercept? e) Now graph the equation. 54. Imagine the graph of the following equation: 4x -5y +20 = 0 What is the slope of the line? 55. The equation C= 0.05t + 10, represents the relations between the total cost (in $) C, charged by an internet service provider, and time (in hours),t, used. a) What is the slope of the equation? b) Explain the meaning of the slope within the context of the equation. c) What is the y-intercept? What does it mean? 56. Student painters charge $5.00 per square metre plus an additional fee of $25.00 to paint a living room. a) Graph showing the relations between the fee charged and the area painted. 2 b) Use your graph to estimate the charge for a living-room which is 25 m . Math Skills for Business- Full Chapters113 57. Kid’s Party Place charges $20 for a party room plus $12 for each person attending. The chart below shows the total cost for 10, 15, and 20 people attending a party. Number in Attendance Total Cost 10 $140 15 $200 20 $260 a) Graph the above information. b) Calculate the slope for the line. c) Write an equation for the line 58. The data below represents the percent growth in the net profit of a company from year 2000 to 2004. Year Percent (%) Growth 2000 20.00 2001 21.50 2002 23.00 2003 24.50 2004 25.00 a) Plot a graph showing the company’s net profit growth. b) Calculate the slope of the line. Math Skills for Business- Full Chapters114 U4 Full Chapter (continued) 12 Systems of Equations 12.1 What is a System of Equations? In section three on equations, you solved equations involving one variable or unknown or placeholder. There are some situations in which two or more variables are involved. Such situations entail two sets of combination of things. However, in this section, we will restrict ourselves only to those involving two variables. A problem that requires two equations to be constructed in order to solve a problem is called a system of two linear equations in two variables. Some mathematicians call this system simultaneous equation because of solving the two equations together rather than separately. The example below illustrates the nature and characteristics of simultaneous equations. Example Suppose you have $75 for shopping. You could buy 2 CDs and 3 blank cassettes with the money. You could also buy 1 CD and 9 blank cassettes with the money. If the cost of CD is d and the cost of a blank cassette is c, find the cost of one CD and the cost of one cassette. Solution Let us construct two separate equations. Since d represents CD and C represents cassette, we have 2d + 3c = 75 Equation 1 1d + 9c =75 Equation 2 We need a pair of numbers that is a solution to both equations. So let us draw a table of numbers for 2d + 3c and 1d + 9c, using different numbers for d and c. We simply substitute each number for d and c in the equations to see if we give exactly $75.00. d 5 7 8 10 20 30 c 12 12 12 6 6 5 2d + 3c= $75 46 50 28 38 58 75 1d + 9c =$75 113 115 62 64 74 75 Math Skills for Business- Full Chapters 115 You will find that in the table there is a pair of numbers for d and c that satisfies both equations. This pair of numbers, d =30 and c= 5 is a solution for both equations simultaneously. Because, 2(30 + 3 (5) = $75.00, and 1(30) + 9(5) = $75.00, that pair is the solution to the two equations. One CD costs $30.00 and one cassette costs $5. We have to repeat the point that to solve a linear equation with two unknowns, two different but related equations are needed. We also have to caution the student that not all simultaneous equations have solutions. When solving any system of equations, you should know that: d) the system may have no solution; e) the system may have one system( one number each variable); f) The system may have infinitely many solutions. The procedure of trial and error that we used to solve the problem in example 1 is not most appropriate. This is because it is time consuming as it involves trying several pairs of numbers until we find the pair that satisfies both equations. As well, the procedure is cumbersome and one is more likely to make many mistakes. We need a better method that is more reliable, simpler, and economical in time. 12.2 Solving a system of Equations Over the years mathematicians have invented three methods for solving a system of equations in two variables. These are elimination (or addition) method, substitution method, and graphical method. The use of any of the first two methods to solve a system of equations depends on the nature of the problem. Some problems are easily solved using elimination rather than substitution. However, the graphical method, despite that it gives only approximated solution could be used to solve any systems of equations. Though three methods or variations of them are customarily used to solve a system of equations, it does not mean that the student should not invent his or her own method for solving such a system. We discuss each of these three formal methods with illustrations. Math Skills for Business- Full Chapters 116 i) Elimination (Addition) Method With this method, one of the variables is eliminated through a process of cancellation. And then the equation is solved for the remaining variable. The number for the remaining variable is substituted in the original equation to solve for the eliminated variable. To eliminate a variable, you must obtain a variable in the other equation that differs from the other in sign. Alternatively, one equation may be subtracted from or added to the other in order to eliminated one variable. Example 1 What is the solution in the form (x, y) for the following system of equation? 4x + 3y = 26 2x – y = 8 Solution In this example the two equations have been constructed already. Our task is to solve it and write the answer in the ordered pair form (x, y). 4x + 3y = 26 Equation (1) 2x – y = 8 Equation (2) If you examine the two equations carefully, you may see that if we multiply each term of equation 2 by 3, we get 2x (3) – 3 = 8 x 3. By adding equation 1 and 2 together are able to eliminate y. 4x + 3y = 26 6x - 3y = 24 Adding these, 10x =50 x = 5 To find y, substitute 5 into x in any of the original equations. 2(5) – y = 8 10- y =8 y =10 -8 = 2. The ordered pair (x, y) = (5, 2) Math Skills for Business- Full Chapters 117 Example 2 Mountain Ridge Restaurant is having lunch special. It is offering 2 slices of pizza and 1 pop (a choice of coke, Pepsi, spirit, C-plus, etc.) for $3.50 or 3 slices of pizza and 2 pops for$5.75. How much is 1 slice of pizza and 1 pop? Solution We have two separate situations and they require two separate equations. Let us take p for pizza and q for pop. Then, we have the following equations to solve. 2p + 1 q = 3.50 Equation (1) 3p + 2q = 5.75 Equation (2) Ask yourself which variable is easier to eliminate? Certainly we can eliminate q, since it is a lone variable in the first equation. To eliminate q, let us multiply the first equation by 2 and subtract the 2 equation from equation 1. We have, 2p + 1 q = 3.50 Equation (1) x 2 4p + 2q = 7.00 3p + 2q = 5.75 Equation (2) 3p + 2q = 5.75 subtracting, p = 1.25 So each slice of pizza costs 1.25. Let us solve for q by substituting 1.25 for p in the original equation. We have, 3p + 2q = 5.75 3(1.25) + 2q =5.75 3.75-3.75 + 2q = 5.75 – 3.75 Add - 3.75 to both sides 2q =2.00 Divide each side by 2 So a pop costs $1.00. Example 3 Eight thousand people attended a Christmas concert in a stadium. The ticket prices were $50 for an adult and $30 for a child under 18 years old. The total revenue from the ticket sales was $250,000. The concert promoters want to know this vital information for the purpose of future planning: How many tickets of each were sold? Math Skills for Business- Full Chapte118 Solution After reading the problem, you will know that there are two categories of attendees at the concert: adults and children. The price for each is different, and the number of tickets sold for each category may also be different. To know how many tickets of each category were sold, we have to construct two equations. Let A be the number of adults who attended, and C be the number of children who attended. (# of adults) + C (# of children) = 8000 (Total number of attendees) A + C = 800 We also know that, (# of adults x $50) + (# of children x 30) = 250000 (Total receipts) 50A + 30 C = 250000 Thus we have to two sets of equations: A + C = 8000 Equation 1 50A + 30C = 250000 Equation 2 We can choose to eliminate any of the variables. Let us say we want to eliminate A. We have to multiply the first equation by a number such that when we subtract equation (2) from equation (1), A will be eliminated and we will be left with only C to solve. Let us multiply equation (1) by 50 and subtract equation (2) from it. We have, 50A + 50C = 400000 50A + 30C =250000 20C =150000 Result for subtracting equation (2) from equation (1). C =7500 Result for dividing both sides by 20. A total of 7500 children attended. Since 8000 attended we know that 500 adults attended (8000 – 7500). So 7500 children and 500 adults attended. Math Skills for Business- Full Chapters 119 Example 4 Redco Inc. Manufactures both 8” portab
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