POH103 Linear equations 2023

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Oct 30, 2023

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Linear Equations and Graphs POH103 Data Management Developed by: Ian Young, Associate Professor Updated by: Aida Haghighi, Assistant Professor School of Occupational and Public Health Toronto Metropolitan University
At the end of this lesson, you were able to: Interpret, evaluate, and solve various mathematical problems related to algebra Objectives of the previous lesson
At the end of this lesson, you will be able to: Interpret, evaluate, and solve various mathematical problems related to equations Discuss the applications of linear equations and how they can be applied to different real-world scenarios Objectives of today’s lesson
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1. Cartesian Coordinate System 2. Graphing Equations 3. Slope 4. Parallel and Perpendicular Lines 5. Applications 6. Exercises and Poll Everywhere Quiz Questions 7. Homework Lesson Plan
Cartesian Coordinate System Cartesian coordinate system = a grid system having perpendicular axes Based on a two-dimensional plane consisting of the x-axis (horizontal line) and the y-axis (vertical line) The x-axis and y-axis are perpendicular to each other, dividing the place into four sections, called quadrants Quadrants are numbered counter- clockwise Origin = the centre of the plane, where the two axes cross, i.e. point (0, 0) From the origin, each axis is further divided into equal units Image source: Abramson et al., 2015
Cartesian Coordinate System The x-axis increases on the right and decreases on the left, while the y-axis increases up and decreases down Both axes extend to infinity Each point in the plane is identified by its: x- coordinate = horizontal displacement from the origin y- coordinate = vertical displacement from the origin Together, we write these numbers as an ordered pair ( x, y ) Image source: Abramson et al., 2015
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Cartesian Coordinate System Example: we can represent the point (3, −1) in the plane by moving three units to the right horizontally, and 1 unit down vertically Image source: Abramson et al., 2015 When dividing the axes into increments, we can consider the x- axis separately from the y- axis I.e. the axes may represent different number scales and different units
Exercise Plot the following points on the plane: (−2, 4) (3, 3) (0, −3)
Question The ordered pair of (−3, 2) would be in what quadrant? A. I B. II C. III D. IV
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Graphing Equations Equations whose graphs are straight lines are called linear equations The only power of the variables is 1 E.g. 2? − 3? = 6 2? = 3 To graph a linear equation with two variables, we need to find the x and y coordinates of a set of points We can accomplish this by: Selecting at least three arbitrary values for x or y Make a table to list the x or y values, equation, and coordinates For each selected value of x or y , solve for the other variable Plot the points on the Cartesian plane
Example To graph the equation ? = 2? − 1 : ? ? = ?? − ? (?, ?) −2 ? = 2 −2 − 1 = −5 (−2, −5) 0 ? = 2 0 − 1 = −1 (0, −1) 2 ? = 2 2 − 1 = 3 (2, 3)
Exercise Graph the equation ? = −? + 2 ? ? = −? + ? (?, ?) −3 0 3
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Graphing Equations The points at which the graph crosses the axes are called the intercepts x -intercept = point where the graph crosses the x -axis. At this point, the y- coordinate is 0 y -intercept = point where the graph crosses the y -axis. At this point, the x- coordinate is 0 To determine the intercepts, set one value ( x or y ) to 0, and solve for the other Example: ? = 3? − 5 Intercept ? = ?? − ? (?, ?) ? = 0 ? = 0 − 5 = −5 (0, −5) ? = 0 3? = ? + 5 ? = 0 + 5 3 = 5 3 ( 5 3 , 0) x -intercept y -intercept
Question What are the y -intercept coordinates of the following equation? 1 4 ? = 4? + 4 A. (0, 16) B. 16, 0 C. 1, 16 D. (0, −1) E. (−1, 0)
Exercise Find the intercepts of the following equation: 4? = − 1 2 ? + 1 Intercept ?? = − ? ? ? + ? (?, ?) ? = 0 ? = 0
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Graphing Equations When the equation of a line has only one variable, the graph is a horizontal or vertical line If y equals any value, the line is horizontal, if x equals any value, the line is vertical E.g. ? = 3 E.g. ? = −2
Exercise Which of the following graphs corresponds to the following equation? ? = −5 + 1 2 A C B D
Slope The slope of a straight line is the rate of change in y over x (i.e. rise over run) Slope refers to the ‘steepness’ or inclination of a line, as well as whether it rises or falls Mathematical expression: 𝑚 = ? 2 − ? 1 ? 2 − ? 1 When a linear equation is expressed in the slope-intercept form : ? = 𝑚? + 𝑏 If the slope is positive, the line slants to the right if negative, the line slants to the left As the slope increases, the line becomes steeper The slope of a horizontal line is 0, and the slope of a vertical line is undefined
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Example The slope of a line that passes through the two points 2, −1 and −5, 3 : 𝑚 = ? 2 − ? 1 ? 2 − ? 1 𝑚 = 3 − (−1) −5 − 2 = 4 −7 = − 4 7 x 2 y 2 y 1 x 1
Question What is the slope of a line that passes through the two points −3, −2 and 7, 1 ? A. 5 4 B. 1 4 C. 1 10 D. 1 6 E. 3 10
Exercise Graph the line that passes through the point −1, 4 and has a slope of 2 3
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Exercise What is the slope of the following line: 2? = 1 2 ? + 4 A. 1 4 B. 1 4 C. 4 D. −4 E. 1 2
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Exercise What is the equation of a line that passes through the point 10, −5 and has a slope of 2 5 ? A. ? = 2 5 ? + 12 B. ? = 2 5 ? − 1 C. ? = −5? + 2 5 D. ? = 2 5 ? − 9 E. ? = 5? + 2 5
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Slope The point-slope formula can also be used to solve linear equations: ? − ? 1 = 𝑚(? − ? 1 ) The equation can then be expressed in either the slope-intercept form (? = 𝒎? + 𝒃) or the standard form (?? + ?? = ?) The slope of the standard form equation = ? ? Example: Use the point-slope formula to find the standard form of an equation that passes through point −1, 3 and has a slope of −2 ? − 3 = −2(? − −1 ) ? − 3 = −2? − 2 2? + ? = 1
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Exercise Use the point-slope formula to find the equation of a line that passes through the points 3, 4 and 0, −3 . Write the answer in the slope-intercept format.
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Question Use the point-slope formula to find the equation of a line that passes through 1 4 , −2 and has a slope of −6 . What is the answer in standard form, without any fractional terms? A. 24? + 4? = −1 B. 24? + 4? = 1 C. 12? + 2? = −1 D. 12? + 2? = 1 E. 12? + 4? = 1
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Exercise What is the slope of the following linear equation? 6? − 4? = 7 A. 2 B. 10 C. 1 7 D. 3 2 E. 3 2
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Parallel and Perpendicular Lines Two lines are parallel if they have equal slopes: 𝑚 1 = 𝑚 2 Two lines are perpendicular if they form a right angle (90 degree angle) when they meet In this case, the slopes will equal -1 when multiplied together, so: 𝑚 2 = − 1 𝑚 1 Example: find a line parallel to ? = 5? − 3 that passes through the point (3, 1) We know the slope is the same, so we substitute points into the equation (y=5x+b) and solve for the constant 𝑏 : 1 = 5 3 + 𝑏 𝑏 = 1 − 15 = −14 Therefore, the parallel line equation is: ? = 5? − 14
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Exercise What is the equation of a line that passes through the point (0, 5) and is parallel to 8? − 2? = 5 ? A. ? = 4? − 5 B. ? = 4? + 20 C. ? = 8? − 25 D. ? = 8? − 5 E. ? = 4? + 5
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Question What is the equation of a line that passes through the point (4, 4) and is perpendicular to ? = 2? − 12 ? A. ? = 2? − 6 B. ? = − 1 2 ? − 6 C. ? = 1 2 ? + 6 D. ? = − 1 2 ? + 6 E. ? = 2? + 6
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Applications Linear equations can be used to fit many real-world applications To solve these, we need to first determine the known quantities and define the unknown quantities as variables We then interpret the words as mathematical expressions Example: A workplace health and safety consultant charges $25 per participant to run an on-site training course, plus a $1500 flat fee. What is the cost to train ? workers? What would be the cost to train 120 workers? The cost of training 120 workers = ? = 25 120 + 1500 = $4500 The cost of training ? workers = ? = 25? + 1500 ? represents the number of workers
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Example If it costs $1125 to vaccinate 17 people against MMR, and it costs $2250 to vaccinate 62 people, assuming a linear relationship, how much would it cost to vaccinate 100 people? A. $3000 B. $3200 C. $3350 D. $3375 E. $3500 𝑚 = 2250 − 1125 62 − 17 = 1125 45 = 25 Substituting with 17, 1125 : 1125 = 25(17) + 𝑏 𝑏 = 1125 − 25 17 = 700 Find the cost for 100 people: ? = 25 100 + 700 Let ? = the number of people, and ? = the cost ? = $3200
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Exercise Nitrogen oxides (NO x ) are an important cause of air pollution in Ontario. In Toronto, the average levels of NO x have fallen from 24.2 parts per billion (ppb) in 2007 to 15.9 in 2016. Assuming a linear relationship, what would the levels of NO x be in 2025? Round your intermediate calculations to 3 significant digits, and your final answer to 2 significant digits.
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Applications Finding the intersection of two lines: At the intersection point, the x and y values for both lines are the same To find this point, we let either the x or y values be equal Example: Find the intersection of the following two lines: ? = 3? + 1 ? = 6? − 5 6? − 5 = 3? + 1 6? − 3? = 5 + 1 3? = 6 → ? = 2 Substituting ? = 2 into either equation gives us ? = 7 Hence, the intersection is (2,7)
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Exercise What is the intersection of the following two lines? A. (−3, 3) B. ( 7 2 , 15 2 ) C. (3, −3) D. 3, 5 E. 5, 3 ? = 5? − 10 ? = 1 3 ? + 4
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Applications We can also find the intersection of two lines using the elimination method: We eliminate one of the two variables by adding the left and right sides of the equation together By answering the following questions. If the answers to the questions are yes, the left and right sides are added together: 1. Are the numbers in front of x (or y) the same? If not, we need to make them the same 2. Considering the variable selected in question 1, is one of them positive in one equation and negative in the other one? If not, we need to switch the signs And then, we add both sides of the equation to get rid of one of the variables and solve for another variable(remaining variable).
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Applications Example: Find the intersection of the following two lines: 2? + 2? = 8 3? − 2? = 2 2? + 2? = 8 3? − 2? = 2 + 5? = 10 ? = 2 2(2) + 2? = 8 ? = 2 The intersection is (2, 2)
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Question What is the intersection of the following two lines? 4? − 2? = 6 −5? + 3? = 1
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Exercise You are considering two consulting companies to improve workplace health and safety at your firm. Company A charges $35/hour and a $2500 flat fee, while Company B charges $40/hour and a $1750 flat fee. If you require 50 hours of consulting, which company offers the better rate? A. Company A B. Company B
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Exercise You are considering two consulting companies to improve workplace health and safety at your firm. Company A charges $35/hour and a $2500 flat fee, while Company B charges $40/hour and a $1750 flat fee. What is the break-even point in the number of hours that would yield equal value from both companies?
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Homework Read Chapter 2 in textbook, complete practice questions Some of the practice questions will be reviewed in next week’s tutorial session.
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Conclusions You are able to: Interpret, evaluate, and solve various mathematical problems related to equations Discuss the applications of linear equations and how they can be applied to different real-world scenarios
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