Introduction to Linear Relations Example 1 : The cost of math tutoring varies directly with time, in hours. Ella charge s $20/h. Let y represent the amount she is making Let x represent the hours she tutors a) Create a table of values b) Graph the relationship c) Write an equation to represent this situation y, amount Ella changes depends on x, number of she tutors. d) How much will Ella make if she tutored for 20 hours? Sub. x=20 in y=20x
𝑦 = 20𝑥 = 20 × 20 = 400 Ella will make $400 if she tutored for 20 hours. Direct Variation • occurs whenthe dependent variable varies by the same factor as the independent variable •can be defined algebraically as y = kx or y = mx is a straight line that passes through the origin y=kx 𝑦𝑥 = 𝑘𝑥 𝑥 𝑦𝑥 = 𝑘 𝑘 = 𝑦𝑥 OR 𝑦 = 𝑚𝑥 𝑦𝑥 = 𝑚𝑥 𝑥 𝑦𝑥 = 𝑚 𝑚 = 𝑦𝑥 Note: x is the independed variable and y is the dep. variable. k or m is called the constant of variation. Example 2 : Determine the constant of variation for each of the following. a) The distance travelled by a car varies directly with time. The bus travels 300 km in 4 hours. b) y varies directly with x. If y=70 and x=10
Solution: a) Distance is the dependent variable Time is the independent variable m=(Distance)/(time) =300/4 =75 b) y depends on x So, m=y/x =70/10 =7 Example 3 : Consider the graphs of d = 2t and d = 3t. a) Describe the similarities In both graphs, the line goes through the origin and the lines rise from left toright. Note that of depends directly on t. b) Describe the differences. Explain why these differences occur. One line is steeper than the other. The equations have different constants of variation (2 and 3). d=2t and d=3t m=2 m=3 Example 4 : Which of the following would you classify as direct variation ?
The line does not passthrough the origin. This is a curve which doesnot go through the origin. Direct variation since theline goes through theoriginal.